I had a similar issue with my previous rig. I had a crappy PSU, and a Geforce 570 and the system would restart randomly. I sold that computer to my brother and the issue persisted. I let him have an old Geforce 450 I had lying around and the restart issue stopped with the new vid card, but I don't know if I could say that proves that the issue isn't related to the PSU... after all, a 450 draws less power than a 570 which would alleviate stress on the PSU. If I had to choose, I would suggest a new vid card but it sounds like you could probably use a better PSU as well.
I know my brother wasn't convinced that the 570 was bad and he was planning to apply new thermal paste to the GPU to see if that has any affect.
You're going to want a good PSU eventually, though. A crappy PSU can cause issues that can totally ruin a systems performance and stability. I bought many of my computer's components used/refurbished, but I bought a brand new evga 750W power supply because it's the foundation of a PC. If your PSU sucks than your entire system potentially sucks.