This is my personal opinion but talk with your parents/guardian/trusted adult about what the best thing to do would be (legally speaking). While there might be something wrong with the system itself like information recorder wrongly or whatever, they might also be trying to cheat you out of your money because they thing you are young and dumb. This is very much illegal and could have very bad consequences for them. I would say that go with a legal adult to confront manager (in person would be preferable because they would feel pressured at the presence of someone with more life/work experience. Demand that the situation is resolved whether it is a problem from the system or not. If it does not work, getting legal assistance from a lawyer might be your best bet, I guess.

Good luck with everything.

So, unless you set up direct deposit your paycheck was likely either mailed to you (and hasn't arrived) or it's at the store you worked at. If this is NOT the case and you straight up haven't been paid for your shifts then you don't have much choice but to go in and ask for it. If the store manager continues to be a problem and doesn't pay you it becomes a labour law issue and you'll have to escalate it above the store level.

Aside from that, if you haven't been officially terminated yet for missing work you'll have to submit your resignation in writing, but if you only worked a month and a bit you might have been on probation which can negate some of the rules around being fired/quitting, so again do your due diligence and find out if it's required. I don't think you'd have qualified for benefits and won't have a lot of things to claim/close up with Starbucks either, but if you end up needing to claim unemployment in the future or anything along those lines you'll probably need to have proof that you were employed there at one time and how long you were there, your reasons for leaving, etc.

Unfortunately you will have to communicate with your store in one way or another if you want to tie up the loose ends. There isn't much getting around that unless you get someone to be a go between. You could also phone or email your store supervisor instead.

EDIT!! just realized I misread your original post about the earnings and tbh if you're not owed any money by them you could probably just move on. Paychecks are for every two weeks and if you haven't been there then you haven't earned anything to be worried about. Like I mentioned, check if you were still probationary because that makes it a grace period where you can usually leave without two weeks notice without much in the way of consequences. It's your first job so it won't follow you around the rest of your life and I wouldn't even bother to put it on your resume if that's something you're worried about. You could still email them if you wanted just to say that you're not coming back but that's about it. You don't really owe them anything as far as I can tell.