That's a mediocre mobo. And the GTX 560 is woefully underpowered. The HD 6870 beats in singlehandedly for about $130-$140. The i5 3570k is only $50 more, and you get $50 off your motherboard, so you get a better CPU and motherboard for the same price.
Is the HD 6870 the best card for money? I chose the GTX 560 based on newegg rating. I don't enjoy overclocking and was wondering if the i3-2120 was a better idea for my purposes of just checking out new games. In short: If I modified the build to have the HD 6870 and i3-2120 would this still suit my purposes? Or is the i5 3570k still advisable despite the lack of interest to overclock?
The i5 is a lot better. It's quad-core, faster, Ivy Bridge, supports PCI-E 3.0/SATA III/USB 3.0, etc. When your computer starts to bottleneck(not in the forseeable future), you can overclock to overcome the bottlneck. It's very easy, and all it needs is a CPU cooler. The CoolerMaster Hyper 212+ is $20 and very good and you'll go up to 4.5GHz. I recommend it above all else. If you want to save $20, get the Sandy Bridge version (2500K) which is slightly slower. However, i recommend the 3570k as you will not spend extra over the cost of the i3. Some games like Skyrim are CPU hogs, and they need that quad-core power. Skyrim with the hi-res texture pack needs at minimum a quad-core CPU, and in the next few years, most games will fully use quad-core CPU's. The 6870 can max BF3 on Ultra at 60FPS. The 560 is slower. The 6870 also overclocks better. Heck, it performs better and it's cheaper! I myself am getting the 6870 and i5 3570K this summer to build my own rig.