Just to be clear because you didn't state it in your post, you must unlock your phone prior to using a non-Verizon SIM. You can sometimes get these unlock codes from Verizon but you can also buy them online. I've used GSMLiberty.net with success before. After that, you should be able to pop in your SIM from the new carrier and begin using it right away. In some cases, you may need to manually add the correct APN for your carrier. Find these settings under Settings > More networks > Mobile networks > Access point names. Click the little plus icon in the top right and manually enter AT&T's APN information. You may need to Google that as I don't know AT&T's APN info.
I use my Verizon S5 on T-Mobile's networks with absolutely no issues, I assume it would be a similar experience for AT&T as well. For me, I put my T-Mobile SIM card in and could make calls and receive texts, but mobile data and MMS was not working properly until I manually added the correct APN information. After that, MMS, 4G LTE data and everything is working flawlessly.
