i used to dread doing this, but as long as the new pc has an IDE connection in the motherboard then you should be ok.
if you have the make and model of the drive then you have to go onto the manufacturers website to find out how to set the drive (old one) to Slave mode so your OS doesnt load up along with your existing one.
Take the old drive, set the jumpers to slave via the back of the drive where the 6 very small pins are. Plug in the IDE (grey ribbon cable) cable to the IDE connection on the motherboard normally called IDE1, cause your optical drives are in IDE2 and take a power adapter from the PSU, make sure the drive is in the case securely and start it up. Your BIOS should recognise that its a slave and boot your OS from the good drive and you should be able to access your old drive as an extra Drive in your PC then just transfer the files accross manually.
Hope that helps