I use Milestones for leveling my current game, and I'm kind of stuck trying to figure out when to move the team to Level 3. We're approaching Session 5 tomorrow, in which they'll likely defeat the boss of their first dungeon and, with a little luck on the time side of things, they'll get a lot of answers about the shadowy things happening in the town and out in the dark woods; lore and information that will define the rest of this first adventure.

Story-wise it would be good to let them level up to Level 3 after this session; my issue is that within the game, it's been barely two whole days since the game begun. It feels strange to me to let them level up so quickly ingame, while it feels kind of natural as a "now the training wheels are off"-moment after they uncover some truths about the issues plaguing the city they're in.

I leveled them up after a pretty slow second Session of investigation and talking to NPCs, just because I felt like it would feel better for them to have a little extra oomph to their character skills.

What's your take on this?

I think that if you're going to use story-based advancement, then go with the story beats that make the most sense for leveling up.

Time that's passed is not traditionally the limiter on when Characters advance in level - it's how much they've accomplished. Or not. You could just fire the level-up on the appropriate story points - but it's sounds like you want to preserve some sort of relationship between "things accomplished" and "when they level". 

So - I'd suggest moving the goal posts. Still have them level up when they defeat the boss of the first dungeon ( I agree that's a natural story beat for that to happen ), but stuff more material between them and the chapter-end show down. You don't have to make more time pass in the game world - just make it so there are more obstacles and challenges between them and the chapter end that they have to overcome. That will likely move them leveling up to a later session than you'd predicted, but they will have accomplished more to get to the natural level-up point of the story, so a level-up will feel more earned.

Good luck!

Vedexent, as usual, is right. I’d just add that the first three levels are always pretty quick. A lot of character classes don’t even truly start until level 3, so there’s nothing wrong with getting there quickly and then slowing things down a bit. 

My players gained level 2 on the 2nd in-game day. They had one fight the first day, that took a lot of spells out of them (given how few they have) and although they wanted to follow up, they were afraid they would need spells, so they waited overnight and hit the cemetery (vs. undead) the next day. Had they gone the first day, they'd probably have survived, and they'd have hit level 2 on the first day of their existence.

I see no problem with this. You will find that sometimes long swaths of time will go by (travel) with 0 leveling. Other times, in a few hours of character time a lot will happen and they will level. That's just how it is. It happens in stories as well. Sometimes an author can dismiss weeks or months with a single sentence and then spend 3 chapters on 20 minutes of time. This type of time compression is necessary in stories. I wouldn't let it bother you that they hit level 3 in a couple of days. My party will probably hit level 3 on their 4th day (2 days after level 2)... so it's not that much different. It doesn't seem "too fast" in the context that it will be after a total of about 6 sessions.

I like milestones since it relates so well to narrative.  Just ask yourself: "Did the party accomplish something significant?"  
There will always be certain players who ask incessantly about gaining the next level.  Just stay focused on how the story unfolds, and don't worry too much about time intervals.  

J.R.R. Tolkien said “Now it is a strange thing, but things that are good to have and days that are good to spend are soon told about, and not much to listen to; while things that are uncomfortable, palpitating, and even gruesome, may make a good tale, and take a deal of telling anyway.”

Story telling in a game is like this as well.  Characters may have spent only hours or days in a dungeon crawl and weeks traveling along the High Road with a caravan, but it is the former that is going to give them more meaningful experience.

I have more or less exclusively used milestone progression for my parties as well. I am a newer DM, but for me as I am outlining things I kind of have an idea of where the milestone leveling will be as part of my story narrative. It helps me create kind of succinct points (such as end of a tough antagonist, or end of an arc of the story, or end of a characters progression, etc). Then when I kind of have that goal post in mind its easier for me to improvise and add in encounters and skill challenges based on my characters decisions because I know where their next 'milestone' will be. It also helps me know fi I need to flush out some things if I have two milestones too close together and need to add or expand as needed.

That being said I wouldn't feel bad about having people get to level 3 quickly. As has been said previously a lot of classes don't even begin until level 3, and if you were going from an official module most of those stories don't even kick in until level 3. You want your folks to get there quickly!

Thanks for the advice, they'll get a level within the next two levels then!

Also, good idea with moving the goal post Vedexent, but we literally ended the last session with them looking into the main room of the dungeon with about a dozen red eyes looking on them from the darkness so it's not an option this time :D

And gotta say, I'm stoked. The poor madmen rescued a teenager who tried to brave the sewers and find answers alone, but she was way in over her head... And they decide to take her with them instead of leading her out, and now they're about to fight a nest of ghouls. Bless their sweet souls, they've got no idea.