I am looking at building my first gaming PC, on a budget that allows $500 on the computer itself (not including peripherals), and am currently researching which graphics card will best suit my needs. As it stands, I do not play any FPS or MMO games at all; meaning no Crysis, WoW or Battlefield. Basically, I only play single-player RPGs, and other similar story-driven games. The newest games I would playing as of this posting would be Skyrim and Mass Effect 3, and I would like to be able to enjoy them at fairly high settings.
The best recommendation I can give is an Intel i3-2120 + Radeon 6850. The 6850 is about 80 dollars more than the 6670 but it will mop the floor with Skyrim at 1080p with HD textures installed. It's probably the best bang for your buck right now.
500 dollars is a bit limiting. If you can come up with another $100 to $200 or so you'll be able to get something that will last longer and be more enjoyable. Keep in mind that this is something that you're probably going to have for several years. If you keep the PC for 4 years, then $200 dollars now is just $50 per year when amortized.