Any advice for asking out a cashier at her work? Is that weird and stalkery? How do I avoid embarrassing her in front of other customers, etc.? I'm not good at this stuff!
Don't ask her out without establishing a connection first. That's putting the cart before the horse, and most girls won't give their number out to a total stranger. So flirt and establish some kind of chemistry first.
If you go there semi-regularly you can roleplay that she's stalking you because every time you go there, she's following you. After a couple visits and you calling her out playfully each time, she's now stalking you, and you can transition into drinks or an invite to something else. At that point, it's natural that she wants to see you because you've developed that bond.
It shouldn't matter whether there are other customers there or not. A little flirting isn't going to bother anyone. That's another thing, you should be flirting with everyone, even people you're not interested in. When that becomes part of who you are, it's natural to flirt with a girl you are attracted to and the actions come naturally.