FYI this has been covered in several other threads, so if you need more details you might search for them.
In a nutshell though: The rMini hardware is identical regardless of carrier and from a strictly technical standpoint you can use any carrier with any cellular rMini. Any barriers are due to the carriers. Generally speaking, Sprint will only let you use a Sprint rMini and Verizon will only let you use a Verizon rMini.
You can use any cellular rMini on AT&T and TMobile; just need a SIM. Order for free online for TMobile (threads pop up when there's a freebie deal) or walk into a corporate AT&T store and ask for one (gave me one free).
So... buying an rMini I'd order a Verizon version so I could use it with Verizon, TMobile, or AT&T as I wished.