Buying a low-end CPU is going to give low-end performance. An Intel 4-core CPU is often 50-100% faster than an AMD 4-core, which is why it's priced twice as high. However, you may find that performance is still adequate for your expectations. What's more important is what CPUs are a better value in each price bracket.
Most will probably agree that under $100, AMD's 4-core chips are the way to go. There's an argument to be made for Intel's Pentium G3258, which can be overclocked, but often games just don't run well on dual cores these days, despite those cores being very significantly faster than AMD's. You might consider an
Athlon 860K for around $75.
Moving up to ~$100-200, you have the Core i3 ($120ish), FX-83xx ($150ish) and Core i5 ($190ish). The FX has 8 cores while the i3 is only a dual core with hyperthreading, but in a vast majority of cases, games run better on an i3 than an FX because games rarely make effective use of more than 3-4 cores and the i3's cores are so much faster. An i5 (4 cores, no hyperthreading) is often considered the lowest-end CPU for a "no-compromises" experience in games; that is, you will rarely, if ever, be bottlenecked by your CPU.
Above $200 you can pick up an unlocked i5, which will give another maybe 20% performance from overclocking. AMD has some FX chips in this price range but they're very rarely recommended, because they deliver similar or worse performance to Intel's offerings while drawing significantly more power, producing significantly more heat, and running on very antiquated motherboards/chipsets with older featuresets.
Here is an example of a game which makes good use of AMD's many cores:
Here's an example of a game which does not make use of more than a few cores:
EDIT: Personally, I value efficiency and acoustics, so I built my wife's gaming machine with an i3.