The kind Sheila E. played? It’s two snare drums on a stand, one or more cymbals, and a cowbell or two. I’ve also seen the same setup used by a member of the Miami Sound Machine.
If I wanted one of these setups, could just go to the music store and say, “I want a _______” or would I have to piece it together myself?
Generically, you’re talking about Afro-Cuban percussion. This site even has a “Sheila E” line of drums (along with a variety of instruments from other traditions):