Processors do make a difference. I used to use the AMD Phenom II 965 that AWAR is recommending. Buying that will definitely hold you back. I had a AMD 6950 graphics card; with that processor, I got 25-40 FPS in Sleeping Dogs. Usually around 35, it played a bit rough. When I built a new system with a i7 3770k in it, I continued to use my 6950 while I waited for my new card to come in. Sleeping Dogs shot up to a very solid 60, with almost no dips. The 965 does run some games very well, but it definitely will limit your GPU's potential (don't think it'd matter with a AMD 6670).
If you want to play MMOs you definitely shouldn't get a Phenom II 965. It ran TERA and Guild Wars 2 badly enough that it bothered me, dropped to low 20s all the time. If you're an MMO player, you'll want a good CPU.
Get an i3 3220 for $130 or jump up to the best i5 you can afford. Once you reach the i5 price range, Intel is much, much better. It may be more expensive, but AMD is just bad at that level and up.
From what I've read: i3 3220 beats AMD FX 6300 in most games; in some cases the FX 6300 does better, mostly in newer games like Crysis 3. However, the FX would be a better buy if you use programs that benefit from having more threads, such as video or photo editing--FX's per thread performance is much worse than Intel's, but the FX has more threads than the i3. At $180+ range, Intel destroys AMD.
Maybe you should buy a solid foundation, and then get a graphics card after you save for a few months? If you can wait, you don't have to buy the whole system at once.
Here are two of the best resources I've found on the internet for learning about building your own PC:
I need a new PC! thread on Neogaf [http://www.neogaf.com/forum/showthread.php?t=600246]
Falcon Guide [http://s1002.photobucket.com/user/The_FalconO6/media/CurrentLogicalPCBuyingGuide/Guide.png.html]