What games do you play? What do you want to play?
My desktop was similar to yours in many ways (GTX 650Ti, Core 2 Duo 3.0GHz, 4GB RAM).
For $300, I got a new MOBO (LGA1151), DDR4 memory, new processor (a G4400 -- the slowest skylake processor on the market -- $65 -- still beats the pants off my Core2 and yours too --
http://cpuboss.com/cpus/Intel-Pentium-G4400-vs-Intel-Core2-Quad-Q6600 )
So for about $400, you could upgrade your MOBO, processor, RAM, and GPU.
But if your PC is that old, I'd recommend a new PSU too.
At that point, you're hitting "buy a new PC" territory.
All of the old Core2 style processors and RAM is very expensive now, so ignore upgrading your system. Start looking at something newer and flashier.
If you can put it together yourself, NewEgg.com has lots of good deals with good reviews. That's where I get most of my stuff. Sometimes it's a bit cheaper elsewhere, though, so it pays too look around.
Also, things go on sale all the time. So don't hesitate to wait. Or... do hesitate. Or something... :) Pick your price point and then look for a great deal that fits in it.
I strongly recommend buying a Skylake processor (LGA1151). All of the new technology is building on that, but it ranges from cheap to expensive. LGA1151 processors range from $65 to about $400. So you can start cheap and then upgrade later. And the MOBOs go from $50 to several hundred dollars, too. So you can get into OCing, or go cheap and stable (my route -- I got the MSI Mortar B150).
Anyway, I hope that helps.
If you have some money sitting around, you may just want to wait until yours dies. Then buy a new PC. If you have a game you want to play now, you might want to do it now.