If you go the building route I would recommend something like this:

PCPartPicker part list / Price breakdown by merchant / Benchmarks

CPU: Intel Core i3-3220 3.3GHz Dual-Core Processor ($119.99 @ NCIX US)
Motherboard: MSI B75MA-E33 Micro ATX LGA1155 Motherboard ($56.98 @ Outlet PC)
Memory: Corsair Vengeance 8GB (2 x 4GB) DDR3-1600 Memory ($47.98 @ Amazon)
Storage: Seagate Barracuda 500GB 3.5" 7200RPM Internal Hard Drive ($54.99 @ Newegg)
Video Card: HIS Radeon HD 7770 1GB Video Card ($99.99 @ Newegg)
Case: Silverstone PS08B (Black) MicroATX Mid Tower Case ($34.90 @ Amazon)
Power Supply: Antec 450W ATX12V Power Supply ($37.73 @ Amazon)
Optical Drive: Lite-On iHAS124-04 DVD/CD Writer ($16.97 @ Outlet PC)
Total: $479.53
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-03-25 09:42 EDT-0400)

CPU: Performance between it and AMD's FX-6300 is quite similar but the i3 will have the upper-hand in older games. I also find Intel builds are a bit more forgiving for a new builder.

Video Card: Your budget is quite limited but a Radeon 7770 is really good for the price right now and should perform way better than anything you've dealt with before.

You're going to have to "download" Windows 7 though if your budget is only $500.

Dell offers a similar build for $450 but with no graphics card, has a 1TB (1000GB) HDD, and the power supply is likely lack luster and not good for supporting power hungry graphics cards.

If you want a second opinion and more building advice check out http://reddit.com/r/buildapc Their sidebar has plenty of links and information for new builders.

Once you build a PC for yourself, you'll never go back to buying them pre-made. Building just gets you way more for the money and you learn a valuable skill in the process.