Well I guess your 9600 GT is a PCI-E graphics card? If yes it should be not problem to use it in a new computer.

About the RAM: You need to check out what type of RAM it is

About your CPU: Check out which CPU-socket your mainboard has and then check if there are any quad core CPUs available for that socket.
If yes you only need to install a new CPU and thats about it.
Otherwise you have to look for a new mainboard which supports a new CPU and (maybe) your RAM.
Then you have to replace mainboard and CPU.

What else you might want to check:
If your power supply is sufficient for your new CPU