Graphics card w/ build

I recently built my first PC and was wondering which graphics card I could get away with playing Battlefield 3 without any lag. I'm not too sure how many FPS I will need, but here's my set up:

Coolermaster Storm Sniper Black Case
ASUS P8Z77-V LK Motherboard
Intel i5 3570k Ivy Bridge 3.4 GHZ
Corsair Vengeance 2x4GB DDR3
Seasonic M12II 620W PSU
Coolermaster Hyper 212 EVO

My price range is between $100 - $250. I don't need anything too special, since the computer is primarily used for graphic design, browsing, and occasional gaming.

I've done some research, but usually just catch myself going back and forth between which card to get.
Also, Radeon or GeForce?

Thank you.