I have been saving up for a gaming pc, I sold my xbox just for extra money. I have a build now that I think will work for my needs for a while. I am choosing Intel because I can later upgrade to a 3570k. My budget is $550ish without windows. I have a monitor, mouse, keyboard already. I will mostly be gaming, browsing, and watching HD youtube videos. I will not be overclocking. I would prefer to have a Intel CPU and a Nvidia GPU. Games I will be playing are BF3/4, Gmod, Minecraft (with shaders, Tekkit/FTB, Texturepacks), and other games that I will get in the future.
With this budget, I would really recommend looking into an AMD build, as I would find the i3 a waste of money in the long run. Nvidia's price-performance ratio is also pretty bad at the low-end cards. And this is coming from an Intel fan.
CPU: AMD FX-6300 3.5GHz 6-Core Processor. Newer games are starting to utilize more cores. Having a 6 core will help.
MOBO: Asus M5A78L-M LX PLUS Micro ATX AM3+ Motherboard. It had everything I looked for.
RAM: Mushkin Blackline 8GB (2 x 4GB) DDR3-1600 Memory. Typical 1600mhz ram. 4GB is not enough for gaming. You'll max out a lot of the time.
HDD: Western Digital Caviar Blue 1TB 3.5" 7200RPM Internal Hard Drive. Typical storage.
GPU: MSI Radeon HD 7850 2GB Video Card. A lot of the load will be on the GPU. I used to have a 1GB 7850, and it worked great for Bioshock Infinite and Source Engine games. It's around the performance of a 660 ti, and super cheap.
CASE: Rosewill RANGER-M MicroATX Mini Tower Case. Cheap, should fit everything.
PSU: EVGA 500W 80 PLUS Bronze Certified ATX12V / EPS12V Power Supply. Plenty of power for this build.
OD: Lite-On iHAS124-04 DVD/CD Writer. Cheap DVD drive if you need it.
As far as an intel build, this is what i came up with as far as what you're asking for.
Changed the video card because of the performance+free batman :)
Changed your cpu because duel core will not get you far in the gaming world. And for the few extra bucks it would be, you just as soon get what you want. You said you weren't overclocking so there's no need to get the 3570K.
Increased the RAM to 8GB because like NooB said, 8 is almost getting close to a minimum now.
Changed the PSU for a few bucks cheaper to a 500w(why not eh lol)
But for your budget, I would recommend Tech's suggestion above with an AMD system. Because as of right now, you can't beat AMD's price to performance ratio, especially with a budget build. Hope this helps. :)