The performance of a PC is MAINLY controlled by the CPU you have and how much memory you have.
Other things can make small differences (faster hard disk, up to date motherboard, better grahpics card, re-install Windows etc).
As has been said, putting in more memory can help a lot, and is the single easiest and cheapest thing you can do to speed up a computer.
Of course if you have an old CPU you can put as much memory in as you want but you will always be held back by the old CPU.
Replacing the CPU is not so easy, and if you have an old motherboard you will not be able to put a new CPU in it anyway.
One other option is to buy a new System unit. I have seen these for sale for under �200 and it will speed up everything in one go.
I was in Staples the other day and saw a System unit from E-machines that was about �190 that had Windows pre-installed.
You buy it and use you existing screen, keyboard, speakers,etc.