How do I turn my PC into a gaming system?

My processor is AMD athlon 64 x2 Dual Core processor 4400 + (2 CPU's)
Memory is 3.5gigs ram
Directx 11
Display is Radeon X1300/X1500 series.

Should I just get rid of my pc and get a new one when I can afford it or should I buy parts for it?

What kind of motherboard do you have? It'd let everyone know just how much you can upgrade it.. But I honestly think you should get a whole new computer. I can't see your motherboard being able to take any Quad-Core (or more) processors. Merely a guess atm though.

If you don't know your motherboard-- http://www.crucial.com/systemscanner/ --just download that and it will tell you.

If you do get a new computer you can safe your hard drive, cd-drom, power supply and case and save a few bucks. So all you'd really need is a Motherboard/CPU/GPU.. The power supply is a maybe, It depends on how many watts it is and what kind of connectors it has. Another thing you could let us know.