How quickly can I get promoted?

Hello! I just had my first day at Publix. I was hired as a front service clerk. I don't want to say for how much but I'm sure y'all can imagine. And it's part-time.

I got this job out of desperation, but with taxes, even 25 hours will barely make my rent at halfway.

Do they have full time positions? A cashier was also there today, so I don't know how soon a position would open up for that.

People kept telling me it's a company I can grow with, but in the interviews they said if I got a promotion, it'd be to another store. I take the bus and am worried about making it anywhere else.

Can someone please give me some hope? Or let me down easily.