a computer is never faster than the day you bought it. in fact it only gets slower as the years go by, even if you don't ever use it.
also, as far as internet goes, the only way to speed that up is to call your ISP and pay them for more bandwidth.
if you want to give a speed boost to ANY computer, you can always replace the hard drive with an SSD. they cost a bit more, and are relatively smaller capacity-wise, but offer a marked increase in speed.
however, first you should check which processor (CPU) you have. if you have a Celeron, Turion, Sempron, or Atom CPU, you may as well be saving your money for a new computer. those 4 are THE WORST processors money can buy, and if you want speed, you've failed by buying a computer with them.