If you're looking to upgrade in the future, Intel is your best choice without a doubt; for gaming alone I would recommend Intel too. AMD sockets are pretty much outdated these days whereas Intel provides a stronger single-core performance, hence you will notice a better game-play experience (i.e. higher fps) with Intel. Most games only utilise 4-cores at maximum, whereas Intel will provide 4 strong cores in comparison to AMD, where AMD will provide 6 or 8 cores which are weaker (out of these 6/8 weaker cores, only 4 of them will be utilised for gaming). Therefore it is ideal to have a stronger single-core rather than more cores; hence I would recommend Intel. In addition Intel will allow you to upgrade from a Pentium/i3 to an i5/i7 and receive a major increase in performance, whereas with AMD you don't receive that upgrade option.
Also your budget of $500 USD, does that include a monitor, keyboard, mouse, DVD drive, an operating system (e.g. Windows 7/8), WiFi card or any other peripherals? Please reply back and let us know and I will update the build accordingly; for now, let me know if this is adequate:
PCPartPicker part list /
Price breakdown by merchant
CPU: Intel Celeron G1820 2.7GHz Dual-Core Processor ($47.98 @ Newegg)
Motherboard: ASRock H81M-DGS R2.0 Micro ATX LGA1150 Motherboard ($47.99 @ SuperBiiz)
Memory: G.Skill NS Series 8GB (2 x 4GB) DDR3-1600 Memory ($66.98 @ Newegg)
Storage: Seagate Barracuda 1TB 3.5" 7200RPM Internal Hard Drive ($49.99 @ SuperBiiz)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($31.99 @ Micro Center)
Power Supply: EVGA 500W 80+ Bronze Certified ATX Power Supply ($41.99 @ SuperBiiz)
Optical Drive: Samsung SH-224DB/BEBE DVD/CD Writer ($14.98 @ OutletPC)
Operating System: Microsoft Windows 8.1 - 64-bit (OEM) (64-bit) ($92.00 @ B&H)
Monitor: LG 22M35D 60Hz 22.0" Monitor ($99.99 @ NCIX US)
Keyboard: Microsoft Desktop 400 Wired Standard Keyboard w/Optical Mouse ($14.98 @ Amazon)
Total: $508.87
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-01-08 17:05 EST-0500
Unfortunately it's not a gaming computer at the moment. If you want to turn into a gaming system, you only need to upgrade the processor and install a graphics card; every other component is ready for gaming. I'd recommend upgrading to an Intel i5 processor for gaming and at a GTX 750 Ti or R9 270 graphics card; with those two, you should be able to run any game at 1080p with medium/high graphical setting (closer to high).
At the moment the build will easily handle all your essential needs and will also run old games (most games before 2011 with ease) such as GTA IV. With the i5 and a GPU upgrade as mentioned, you will be able to run any game. The system I have built for you includes the operating system (Windows 8), a 1080p LG monitor, and a keyboard and mouse. The build above also includes everything to build the system to get it working; I have also included quality parts which should last you years to come rather than include poor parts.
If you do not need an OS (Windows 8), a monitor, keyboard and mouse, please reply back and let me know and I will build a better system for your budget.
All the best.