Straight talk sells their own phones. Most of them run on Sprint's network, though there are a couple that run on Verizon's.
They also have 2 BYOP programs. If you bring an Unlocked GSM device, you can get a sim card from either AT&T or T-Mobile and use their GPRS, EDGE, and HSPA network. Neither will let you use their LTE network. For AT&T, your device should be compatible with the 850MHz in order to support their network. For T-Mobile, your phone should support 1900MHz for Edge, and 1700/2100 & 1900MHz for their HSPA network.
They also have a BYOP CDMA program with very strict conditions. It must be a Verizon 3g only phone. Since Verizon hasn't released any 3g only phones in about a two years, that means it will have to be quite an old phone. Bring it to a store, and they will flash the software for you.