For $300, you can go for this:
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: AMD Athlon II X4 750K 3.4GHz Quad-Core Processor ($80.98 @ Newegg)
Motherboard: MSI FM2-A75MA-E35 Micro ATX FM2 Motherboard ($59.99 @ Newegg)
Memory: A-Data XPG Gaming Series 4GB (1 x 4GB) DDR3-1600 Memory ($29.99 @ Newegg)
Video Card: Sapphire Radeon HD 7770 GHz Edition 1GB Video Card ($84.99 @ Newegg)
Case: Antec One ATX Mid Tower Case ($29.99 @ NCIX US)
Power Supply: Corsair CX 430W 80+ Bronze Certified Semi-Modular ATX Power Supply ($29.99 @ Newegg)
Total: $315.93
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-29 00:06 EST-0500)
It's a bit over, but I didn't skimp on any of the parts.
The case and psu are both high quality and will last you a long time, but remember that if you plan to upgrade the gpu in the future, 430w may not be enough.
The 7770 is the best card in its price range.
For the $400 budget:
PCPartPicker part list /
Price breakdown by merchant /
Benchmarks
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($116.97 @ OutletPC)
Motherboard: ASRock 970 PRO3 R2.0 ATX AM3+ Motherboard ($64.99 @ Newegg)
Memory: A-Data XPG Gaming Series 4GB (1 x 4GB) DDR3-1600 Memory ($29.99 @ Newegg)
Video Card: Gigabyte Radeon HD 7790 1GB Video Card ($105.38 @ Newegg)
Case: Antec One ATX Mid Tower Case ($29.99 @ NCIX US)
Power Supply: Corsair CX 430W 80+ Bronze Certified Semi-Modular ATX Power Supply ($29.99 @ Newegg)
Total: $377.31
(Prices include shipping, taxes, and discounts when available.)
(Generated by PCPartPicker 2013-12-29 00:08 EST-0500)
I took the $300 build and upgraded the gpu and processor. The FX-6300 is much stronger than the 750k and since it uses a different socket, you needed a different motherboard. The Pro3 is a decent motherboard but you can't beat it for the price. It enables you to be able to overclock the FX-6300 later on.
The 7790 is a step-up from the 7770. Not a huge performance increase, but it's still a step up.