Question about unlocked phone?

I took my current cell phone swimming on accident last week - and was able to save it almost - by taking it apart and drying it out ASAP. But for some reason - it will no longer charge. I've exhausted my ability to revive that phone - and want to replace it with an ebay cell phone - rather than renewing my contract for a discount phone via ATT Wireless.

Can I buy a standard unlocked phone - and then bring it into ATT and get it up and running - or is there something else I need to know. I don't want to buy a cell phone - and then find out the phone I purchased won't work on my current ATT cell phone service.

Thanks