I’ve played a few times in a group but find myself as a new DM. Session 0 is coming up and I am pretty nervous. I want to do the best job possible and so I’m looking for any tips, hints, things to avoid. Any advice would be appreciated!

If you are creating your characters from scratch, you might want to have 2 session zeroes, or at least plan for it. I had 3 session zeroes because not everyone could make it and character creation takes a long time, especially for beginners. I recommend skipping character creation if you just want to get down and dirty and go on an adventure, but if your players like the idea of creating their own characters, be prepared to set aside extra time for that.

I also recommend running something short for your first campaign that can conclude within a few sessions, both as a tutorial for the players and for you as a GM to test things out. The Lost Mines of Phandelver is relatively short compared to other official adventures, but it is still really long for a first time group in my opinion, so I recommend looking for a homebrew adventure shorter than even that. I recommend watching the first few episodes of Running the Game from Matt Colville on YouTube, and here is the first episode, as he has a really short adventure planned out for you to copy. I also recommend watching the first season of D&D with High School Students (first episode) to get an idea of how a D&D game is like in real life, and feel free to fast forward and skip things.

If somebody forget to bring their dice, you can roll dice using Google by simply typing in 2d20, 4d6, etc. in the search bar. You can use Beyond's dice, but it might slow your browser down.

As for the game itself, first level characters are really fragile, so do not be afraid to send in NPCs to help them out, and try to limit the number of monsters in combat encounters to be no more than the number of PCs.

Another GM trick I like to use is to fudge dice rolls on occasion, but that is mostly in case of emergency to save my party from dying, so do not overuse it or else it might become obvious. You are basically the god of your universe, so do not feel bound by the rules nor dice. GMing is like cooking, the rules and dice are guidelines like how recipes are guidelines.

Also remember that not every encounter has to lead to a combat encounter. Bribing and sweet talking your way out of trouble are just as valid. That being said, combat encounters are pretty fun, but it does not have to result in one side completely killing the other side either, and you can use non-lethal combat encounters, such as bar fights and dueling bosses one-on-one to settle disputes more reasonably. And once your players get into a fight, remind them that running away is totally okay too if things get tough.

Thank you so much!!

Don't be afraid to break immersion and tell your players you don't know something. If they want to go somewhere that you haven't prepared anything for yet, you can just say, "Okay, but just as a heads up, I don't have anything prepared if you do that, so we'll need to take a break so I can prepare something". If you don't know exactly how something works and it's going to take a long time to look it up, just make something up in the moment... tell your players that it's going to work this way right now, and you'll look it up afterward, and whatever they did might not work exactly the same next time.

Also... if you're not good at doing voices, you don't have to do that for your NPCs. It seems like the standard with all the popular game streams and all, but you can just tell your players, "In a thick scottish accent, the Dwarf responds..." and then just speak in your normal voice.

Start small, and don't try to map everything out 5 sessions in advance. You don't need an entire world for your first adventure, you may only need a single village and a bit of surrounding area. If you have a storyline that will take many sessions to deal with, you don't need (or even want) to work out any details beyond a rough outline and the things the players might concern themselves with during this session and the next. They'll probably come up with something you didn't even think of anyway. So don't do too much that doesn't need to be done yet, and use your time to prep what does need to be done - setting the stage, the NPCs the players may run into, and the encounters that might happen. If you're using digital tools, make sure you're comfortable with them and do all the setup work in them well ahead of time. If you're more like me and use pen and paper, do the same: get your writeups and statblocks in order beforehand, so during the session you just have to flip to whatever page you need.

And above all, don't stress it. The DM is important but doesn't have to be perfect, and everyone at the table is equally responsible for making the game fun - the players no less than the DM. Plus, you'll get better. This is the least good you'll ever be, but that's fine. I'll bet you're more than good enough now, and you'll improve with every session. Things will probably go wrong. That happens. Take a breath (really, take a breath - in fact, take five minutes if you have to; there's no rush), don't sweat it, and move on. That's how we learn. Some things you really can't prepare for, but that doesn't mean you can't deal with them when they happen. 

Surprised no one has brought up names yet. I keep a list of 10 - 20 names I like from a random name generator.  If your PCs talk to someone you didn't expect you give them one of the names and replace it on your list between sessions.

Edit: few things derail me faster than making up names on the fly ;-)

Quote from Noksa >>

Surprised no one has brought up names yet. I keep a list of 10 - 20 names I like from a random name generator.  If your PCs talk to someone you didn't expect you give them one of the names and replace it on your list between sessions.

Edit: few things derail me faster than making up names on the fly ;-)

 I have this site open on a separate tab every time I DM: http://www.npcgenerator.com/

Even if you don't use all the options it offers, it can be used to quickly generate a name.

Also... write down any names that you use and a quick note on who they are in your story. I've lost track of a few NPCs that I didn't expect the players to come back to and had to scramble to figure out what their names are.