I'm going to take a different tack (as is my thing) as no doubt most people answering are in the "you've only been there for a week, just leave" mindset.
Let me go through this by point:
I've only worked there for a week
No matter what job I've worked in (from burger chains to senior management) have I ever gone into a new role thinking "I am an expert at this", even if you have experience you still need to learn about how the job works, and that takes time, so 1 week is nothing.
The manager made me cry yesterday after yelling at me when I messed up an order
This is wrong, and it can be difficult in any service industry with peaks and lulls, however it could be that the manager has been watching too much Gordon Ramsey and thinking this is how to drive you on (not realising it's Ramsey's passion that drives his frustrations, it's not just shouting).
Having said that, what have you done to pick up the job, have you just relied on what they've given you training wise, or have you tried approaching the same manager for mentoring? You'd be surprised how asking someone for their advice can change their attitude.
In a place like Starbucks, there's a good chance the manager worked their way up, rather than being recruited in, so they may be a font of good tricks for the job.
My gut feel is that you don't want to just jack this in. I'd say this by the fact you're asking this question here. It's easy enough to quit in the first couple of weeks, you'd never need to mention it in a CV, and as long as you do it professionally, it'll be unlikely to ever come back to haunt you.
I'd however do what I can to make it better, and also be seen by the manager to be keen to improve, and you will likely find you'll get more support.
You'll still be able to quickly judge this, and if it doesn't work out, can then professionally leave safe in the knowledge that you tried your best, and have something positive to talk about in your next interview if asked.
I also end up working an extra hour everyday and am not paid for it
I'd also talk to your manager about this, likely someone is making a mistake, and you can get that worked out.