It's my first week at my first job out of college. I started Monday and didn't have any type of training at all. I've already been assigned to a bunch of ongoing projects and I'm getting several hundred emails a day. There's also so many meeting invites that I have to turn down most of them because I'm already scheduled for another project meeting. My Microsoft Teams is also receiving direct and group messages about every 10 seconds. IT policy is that we cannot close Outlook or Teams or mute our computers, so I've been keeping headphones plugged in to quiet the noise. People are emailing me to ask me a question about the status of something, then following up 5 minutes later when I haven't replied yet, then messaging me to ask if I saw their email, then calling me on Teams. It's so overwhelming. I'm not sure what to do, I literally cannot keep up with all of the communication much less get any work done. What can I do here?
Talk to your lead or your manager.
Your manager or the lead developer should be in a position to help you get more into a flow and rhythm of communicating if you need to, and they'll be available to set expectations and guidelines on what you need to and what you don't need to respond to.
One of the things you don't mention is if the emails are going directly to you or if they're going to several others as a part of a distribution list or a CC chain. If they're not going directly to you, it's probably safe to ignore those.
As for the direct pings on Teams, you're going to want to figure out the best way to get notified about things. Some channels you have to keep an eye on at all times so it makes sense to have all messages notify you. Others, it makes sense to be notified only when you're mentioned. There's settings available in Teams to help with that.
I'm not sure what to do, I literally cannot keep up with all of the communication much less get any work done. What can I do here?
Ask your manager what's expected of you at this stage of your employment.
Ask your manager to prioritize your tasks.
Ask your manager how you should handle the communication.
If I had to hypothesize on this, I would say that most of this is "somebody else's problem". The reason I say that is because a lot of corporate communication goes through mailing lists; your team probably has a mailing list, and all communication goes through the mailing list, and you're on the mailing list so you get all that communication in your inbox.
This is how corporate email culture commonly works: If there is a direct recipient (a single human) fora message, that person's email is usually in the "to" line of the email, usually at the start of that line. If there are other parties who may be interested in adding feedback, those are usually additional "to"s or "CC"s. If there are a lot of recipients and/or the recipients don't need to know who each other are, then "BCC" is used.
This makes it very easy: Read the subject line of each email and maybe the first couple sentences; enough to understand what the email is about. If you or your project are called out specifically in the email, then reply to the part that pertains to you. If you are the only recipient of the email (there are no other "To", "CC", or "BCC"s aside from your direct personal email), then reply to it. In all other cases (likely the vast majority of cases), ignore it.
It's likely the sender is just putting out a blast email to your whole team to get "someone" to reply; if you have nothing useful to say, then let someone who has something more useful to say respond. The sender is not asking for every member of your team to individually give a status report on whatever they need status of, so you don't need to handle it.
If people are targeting you directly and being annoying, you may need to have a chat with them to tell them to back off. Not aggressively, of course, because it's a workplace, but if someone is asking you to do a task and then 10 seconds later asking if it's done yet, you may want to reply to them and just say something like "hey, that task takes more than 10 seconds to do, just chill, I'll let you know when it's ready, I'm on it, I promise". Some people are overeager in this way and need to be told to chill sometimes.
Short answer: Prioritize
You've already noticed you can't do it all. Most people can't, so most of us get into a rhythm of responding to emails and messages that satisfies most people most of the time. Feel free to reorder these, this is just kind of a general starting point based on my experience.
Priority 1: keeping your manager satisfied. Your manager needs to know what you're working on and where you're at. If your manager asks you a question about these things, answer as quickly as possible.
Priority 2: projects you're directly working on or responsible for. Various stakeholders are going to need updates, even if the update is just "my priority is something else right now but I'll try to get it done by XX date." Help set expectations by giving realistic timelines and statuses, and let people (your manager, other stakeholders) know as early as possible if you're going to miss a deadline.
Priority 3: professional development stuff. In many professions, you need to network and make contacts, attend training classes and social-but-professional events. These are important but not urgent; put them on your calendar and then forget about them.
Anything else probably isn't that important (unless I'm forgetting something) and can wait.
There are various guides out there about "how to manage your inbox" but the advice I got that I still follow is basically "half of your email is garbage and can be deleted, 1/3 of it can be taken care of pretty much anytime, which only leaves 1/6 that needs dealt with NOW." Set aside time to do "real work" but you can expect to spend a lot of time responding to messages - as you've noticed, some of these will pile up if you ignore them for too long. Using a status of "busy" can help set expectations that you may not respond right away. You can also have people contact your manager instead of you directly, but this works best when your manager knows what you're working on and helps prioritize your project work.
The other advice I'd give is to turn off notifications for anything that's not crucial, like Teams channels you're not technically part of and don't need to know every little thing that happens. Browse these periodically so you can keep up with things if it's important to your role. Make some mail rules that auto-deletes some stuff or routes it to a de-prioritize folder. Someone that's in your same role may be able to help advise with rules they've learned over time.
Like anything else, this is a skill that can be learned over time with practice.