I would really like to keep it under $400 however if I absolutely must go up to $500 than I suppose I could make an exception.
Yeah, I figured as much. It's really down to how much you want to spend. A $500 PC will play games more than 25% better than a $400 PC, but a $400 PC will still play games very well. A lot of these guys encouraging you to spend more are accustomed to high-resolution, silky-smooth graphics running on hardware that is much better than average. In fact I just finished a bout of Crysis2 (which isn't great, BTW) at 2560x1400 with the high-res textures and DX11 mode, and whilst it looked nice, it was just as enjoyable last week at 1280x720 without the high-res textures, DX11 features or details set to max. Whilst there are exceptions, a lot of new games are still designed to run well on 6-year-old consoles with antiquated hardware. This means that even a $75 graphics card can do a decent job of smooth gameplay at high(ish) details because it's still ten times more powerful than the graphics abilities of an XBox360.
If you're upgrading from an old HP DV4, you'll be more than happy with a $400 system, and by the time you want more, a $100 graphics card upgrade is probably all you'll need to give it another couple of years gaming.
As for the i3 vs G8/G6 Pentiums, there's basically no difference except clockspeed. Most games don't get any tangible benefit from hyperthreading. Some even perform worse! Unless you plan to spend $150 on a graphics card, you'll probably not see a big difference between a G620 and the top-end i3, because the graphics card will always be the limiting factor in today's games.