I need help purchasing a used iPhone

I'm currently an Android user, but it's time to buy a new phone and I've decided to go with an iPhone. But I have a limited budget and I don't want to buy a phone through my wireless provider. So my plan is to buy a used iPhone 5 on eBay, but I before I do, I want to make sure I get a phone I can actually use. My provider is Tmobile, so I'm assuming I just have to search for a "Tmobile iPhone." Is that all there is to it? (I'm also considering switching over to Verizon, in which case I guess I'd just search for a Verizon iPhone.)

Also, when I get my iPhone, can I just pull the sim card from my old phone and put it into my new phone, or is there more to it than that?

Thanks.