I'm a new DM running a homebrew campaign (borrowing from a lot of shorter adventures). In my last session, my players decided to go to the hideout of a crime syndicate and confront them for information. The party had recently been ambushed by this same group and had some seemingly non-important items stolen and wanted to know what the syndicate's plans were.

I tried to set it up as best I could that this was an extremely dangerous plan without completely telling them "no", but they persisted. During the confrontation, I'd also set up ways they could investigate and find some clues to their larger plan without engaging in combat. However, the players instead offered to do a job for the syndicate in exchange for information. 

In hindsight, I shouldn't have done this, but the syndicate sent the party on a "mission" to rob a casino, but in the end, was actually just a diversion for the syndicate to pull off a larger heist. I'd expected once this was obvious it would actually become an escape encounter but the players tried to stay and fight. The result was a near TPK and the party eventually made a dramatic escape.

Unfortunately, after the session the players started getting frustrated with each other for going through with this plan, blaming each other for it being their idea. They were also frustrated with me for allowing them to go down this path.

I've apologized for making an unwinnable scenario and will take their feedback to heart in the future, but how do I get back on the right track in the next session? Should I lob up a few easy encounters to boost morale? Give them a path to revenge against the syndicate? Open to any and all ideas

You could always go the tough love route. Tell them nobody died, don’t be such babies. They made bad choices and got tricked. It happens. As for what’s next, let them tell you.

I’d also say no to the easy encounters; they’ll know what you’re doing, and it will feel like a consolation prize, which it is. 
Characters getting in over their heads should be a chance to move the story along and motivate the characters. It happens in every story, the good guy loses the first fight, so when he wins the second one, victory is that much sweeter. 

I'd say let them stew on this loss for a bit... don't put them immediately on the path for revenge. I'd go for something flashy but fun to take their minds off of it. If they don't want to let it go, maybe have a disgruntled former member of the crime syndicate seek them out and ask for their help finding something that won't put them in direct conflict with the rest of the crime syndicate. Give them the opportunity to find something to empower themselves, maybe build up more allies to take down this crime syndicate in the future and get their revenge.

Thanks, I appreciate the input. I'd told them in the past that I was trying to run something pretty open world so was surprised with their reaction, but I only allowed it because it should play into the larger plot down the road. 

My initial reaction was tough love. Hopefully, some more playtime dampens this loss.

Thanks for the ideas. I think you're right its best not to solve it immediately. Let it stew and play into a bigger story down the line.

I would say there isn't anything per se wrong, but it may show a mismatch of expectations (classic session 0 stuff).

I think that you needed to make in character knowledge MUCH more clear to the players. Somehow the players got the idea that fights were winnable that organizations were weaker, that success in their goals was possible no matter what the circumstances and no matter what actions they chose to take. It is a certain style of D&D where this happens but clearly not the one you were DMing. You also need to let them know that as DM, you adjudicate the interactions of the characters with the game world that you have created. The DM is not a video game quest giver - where whatever they say is the quest that needs to be completed.

Resolving this situation requires you to sit down with the players out of character and chat about the style of game you are running (real world like, realistic consequences - still heroic but the characters choice of actions matters and each choice may have consequences - forseen or otherwise). Make it clear that sometimes the best course of action might be a strategic retreat. However, keep in mind that if you have given the players an objective to rescue someone, protect someone, or do something heroic or honorable - the odds are good the party may choose to die trying since they think success is what they need in this case as their goal.

In your casino situation, you could have described the guards as very powerful - the characters would have no doubt that the 20 (or however many) guards are likely to easily defeat the party. If you want the party to go into a no-win situation and do something reasonable you have to clearly give them the information that the situation may not be winnable in the first place so they can reasonably decide to retreat.

In addition, after seeing the guards, seeing the fight, seeing how unwinnable it was - the DM should suggest to the characters that the criminal organization might be setting them up. You have to tell the players this at some point, up front since otherwise they might think the mission given by the criminal organization was given by the DM for the players to accomplish - not to fail. The key to these things is timing WHEN to give the players this information that their characters figure out. You could give the characters insight checks when the players get the mission - then let them know that the criminal organization doesn't seem to be telling them everything - use that as the foreshadow when you later reveal what the criminal organization was not telling them ...

I'd say for a party of newer players it's fine to tell them the relative difficulty of the encounter they're going into. A character would know what lethal odds look like, even if a player might not yet.

Narrative phrases like "ok you can tell that the casino has heavy security, and you feel a twinge of great fear as they all start to converge on you" or "yeah this guy looks tougher than anyone you've ever fought before, and yes, he has backup" are your friends when you want to hint at the difficulty of an encounter without fully railroading your players. 

That said, you also might wanna have a talk with your players about lethality. Campaigns vary quite a lot between "nobody dies in this movie, they just get really big boo-boos" and "why are you handing me the phb? You need me to look up a rule?" "You're not looking for anything, you're building character."

Basically, if a player doesn't know where your game falls on that spectrum, if they don't know that a poor decision can lead them to an early grave or if this is the kind of game where they can rest assured that all challenges are balanced and level appropriate,  then they're not going to recognize even the most overt hint that death is around the corner if that's not something they've had to think about in game to that point. 

We had a similar issue to this earlier this year. I'll give some backstory and then tell you what I did to help:

One PC has a backstory of being an assassin rogue but doesn't like anyone to know about it. However because of her backstory she knew things about the crime guild that the party had no idea they were actually tracking down. Unfortunately in spite of me telling her that she knew these things and even going as far as saying "You should tell some of the other players what you know" she didn't. It caused some of the party to come very close to getting in with the wrong people and ultimately they ended up killing city guards and helping criminals without knowing. While no one came close to dying, their characters were crushed to know that that they were inadvertently heading down a very very dark path. Players started getting frustrated with each other and ultimately had people walking out of a session almost angry. 

To fix this, I created a channel in our discord server called "Player discussion" and told them that while I have admin access to everything on this server, I will NEVER look in this channel and they are free to talk about whatever they want to talk about. While I build a very strong "the DM is not vs the player" culture at our table, it's much more fun to have planning sessions without the DM knowing. This one act allowed my players to start talking about things without think I was going to "correct" them when they said something slightly wrong. Long story short, (and I still haven't looked at the discussions in that channel) they came back the next session with great new ideas and knowledge of things that each player knew, so that way they had a lot more fun anticipating what the characters they controlled were going to say/do.

SO... my suggestion to you, create a space for your players to plan and talk about what's going on without you being there to "correct/guide" them. The other thing I would suggest is create expectations. Let them know that you're only controlling the world and that if they choose to jump off a cliff without some sort of ability to fly, they're gonna hit the ground. You can't control how much damage it'll do, you only tell them what it is. Here's an excerpt from my ground rules and expectations:

A general rule of gameplay is that the DM will tell you what is going on in the world, you will tell the table what your character is doing within that world. This doesn’t mean that the DM is against the player characters. What it does mean is that the DM knows and understands what is going on within the world where your characters do not. For the most part, as a player, you don’t have much say in what happens in the world. However, primarily through character development, suggestions for the world are always welcome. Do not expect everything to be exactly how you think it should since there’s parts of the world you’re not privy to.

A lot of good suggestions have been made regarding expectations and player communication so I won't dwell there. I will say that the situation has a potentially huge dramatic upside for your game, the players should really hate the syndicate and if they don't (somehow) make sure the syndicate rears its ugly BBEG head again in the future after the PCs have had some cooling off time. Seriously, have them RP some steam off, maybe two of them even get into a (non-lethal) fight. Rather than a "consolation prize" or "easy" adventure come up with something that reminds them of their positive bonds and what they can accomplish when they work well together. Then you can start a take down the syndicate arch which should be epic. I would love to be a fly on the wall of your table when the reconstituted party takes down the syndicate. Please repost when they do. :)

Winning every encounter is not a guarantee in D&D, nor is it necessary. I think you've been pretty generous to your players by not inflicting any losses. Let them recognize that they're not the most powerful thing in the world, but don't let them get too worked up either- losing a fight is not actually that big of a deal all things considered.

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I've apologized for making an unwinnable scenario and will take their feedback to heart in the future, but how do I get back on the right track in the next session? Should I lob up a few easy encounters to boost morale? Give them a path to revenge against the syndicate? Open to any and all ideas

Personally, I don't think this needed an apology. Considering how many outs you gave them, I don't think it was unwinnable. There's more than one way to "win" a combat encounter, and players should learn that early.

Sometimes you win by being the last ones standing. Sometimes, it's by turning the enemy into an ally. Sometimes, by nullifying the threat. And sometimes, you win by running away and learning a lesson about your hubris or lack of strategy. If you ask me, this circumstance is a perfect time to have an honest chat with your players about how combat encounters and NPC motivations are not guaranteed to be formulaic or straightforward.

I think you've gotten the DM-vs-players misconception out of the way, which is always good, but it does dovetail into a question of expectations. At the end of the day, it's your universe; you are allowed to populate it with characters that plan to take advantage of the party. In my opinion, that makes for a better story and a more satisfying victory when the party finally comes out on top.

You're doing just fine, DM. You're listening, communicating, adapting, and crafting an interesting story. You've got this. 

I tell my players during Session 0 that they’re going to run into some encounters that are so easy one of them can win the fight without any help from the others and some encounters that are so hard that they’re almost impossible for the PCs to win. And I do that, some combats don’t even last a full round and some are harder than hard! They have a chance to “say hi” to a red dragon right now and they’re wisely running the other direction as fast as they can!

I appreciate that, thanks.

Personally I have no issue with "unwinnable" scenarios if by win you mean kill all the enemies.  That just isn't the goal of every fight and in some encounters a fight breaking out is a bad outcome/failure state.  The only thing I worry about is if a fight is survivable, if the players can last 3 rounds before being wiped out then its fine. 

Sometimes you do need some easy encounters though or it becomes draining. 

As for the bickering,  I'd try to nip that in the butt quickly it can get quite toxic.  Its a team challenge sometimes you aren't going to make  the perfect move and that's fine.

I don't think you did anything wrong per se. If possible it might help lighten the mood if you introduce some lighthearted elements in the next session.

I totally understand the dilemma of wanting an encounter to be nearly impossible. It is very hard to instil a sense of impossibility in players, without straight-out telling them "this is impossible" or otherwise relying on metagaming. After all, D&D assumes that player characters are heroes doing the exceptional. One way to help facilitate this notion of impossibility is to implement mechanics that help the players understand this.

The DMG suggests that DMs can describe a creature as being bloodied when it has lost half its hit points. When the first PC is knocked out and the DM hasn't described the opponent as being bloodied yet, I can imagine this triggering a strong sense of "oh no" among the players.

If you want to give players a mechanical hint of whether or not an opponent is too tough to fight, you could draft a base mechanic for all PCs inspired by the Fighter (Battle Master) feature Know Your Enemy and the Rogue (Mastermind) feature Insightful Manipulator (be careful not to invalidate a player's choice of subclass). A potential mechanic could look like this:

Know Your Enemy
If you spend at least 1 minute observing or interacting with another creature outside combat, you can learn certain information about its capabilities compared to your own. Make a Wisdom (Insight) check. Depending on your roll, the DM tells you if the creature is your equal, superior, or inferior in the following statistics.
DC 15: An ability score of your choice
DC 20: Armor Class
DC 25: Level (CR)

If the creature is trying to conceal its true strength, the Wisdom (Insight) check is contested by a Charisma (Deception) check. If the creature wins the contest, its statistics appear as it intends. Once you use this feature, you can't do so again until you finish a short rest.

Personally, i don't usually set encounter difficulty based on my player's reactions. I always remind them that my campaign can have a wide range of encounters difficulty, some easy, other hard and even way too much for them to handle and that not all combat must end in death, they can try to parlay, flee, surrender, be captured etc... depending on the creatures involved, the reason for hostilities etc...

Blaming one another for decisions they made because the outcome was not in their favor is not that constructive. The party has to live with their decisions and assume the consequences for anything they do. Blaming the DM to make things happen to them is also not the way to go, we don't play this game to win or loose, but to have fun and live up to challenges. Constructive criticisms are always welcome, but blaming anyone serves no good purposes to me.  

Depending where they go from there and what they do, the next encounter might be as easier, or harder who knows... if you feel making it easier would be good for their moral you are free to do so, but watch out as that could appear to reward their blaming behavior and encourage them to continue to do so next time things don't go their way.

I wonder if it's not about unwinnable fights, but more about the party's place in the world. The choices the party is making here are big, dramatic choices. They want to be big players in the story and feel important. But it turns out they're just the diversion for the real professionals. They're side players in a story about a crime syndicate, and I get the impression they feel a bit slighted by that.

It's harder to do in an open world where the PCs are dropping into events, but I think there's a way to frame the story so that they feel like important heroes even when they're not the strongest people in the room. Try to give them some goals towards a greater purpose that they can achieve even when playing second fiddle to a greater force. Make sure you remember that the story is about their journey, as opposed to a story about your world where they just happen to be characters.