Which parts do I need the most?
Motherboard, CPU, PSU, RAM. You probably can save yourself the harddrive by taking out the one from your old computer. You'll also need a case, you
might be able to just gut out everything from your old computer and put in new stuff, but you'll want to be sure that it has a microATX form factor.
If you want to do lots of gaming, just remember that with $100 you can easily upgrade the $300 build to have a half-decent graphics card.