As far as stores go, if you're not weirded out by the idea of going to a medical supply store (perhaps with "instructions from your mother," as I did when I was younger lol), go there and see what they have. Some of them carry the "exotic" brands like Tena, Attends, or even Molicare once in a great while. Those brands are always going to better than your typical drug store varieties.
If you just want some damn diapers and don't care, Depends overnights are bad, but if you don't know any better, they are padding for your butt. That will get you through.
I wouldn't go with the drug store brands (Walgreens, CVS, etc.) because I've had nothing but problems and disappointments over the years (and very sparse usage) with them.
Depends are a good starter diaper. Don't expect anything terribly babyish, comfortable, absorbent, or soft, though. It'll just be something you can wet a bit in, wear around the house, and be diapered in. Not the first choice of anyone with a choice.
Once you get a credit or debit card, get yourself something nice. Abenas or Bambinos will be like night and day compared with what you can find around your house. Happy hunting.