Basically, you’ll have to remove the hard drive from the old computer, and connect it to the new computer. If you search online, you should be able to find ‘external hard disk casing’, which acts as a sort of adapter from whatever interface the hard drive uses, to a more friendly interface like USB or firewire.
I am no expert on this, and you don’t seem to be one either, and although you couldn’t really do much damage to your data unless you drop your hard drive or something, I would recommend calling in an expert, as that would make the task quick and easy, and also they might be able to use their own hardware, saving you some hardware costs. For example, if you don’t wanna use the old hard drive later, then the external casing will just be used once and wasted. When I shifted to my new computer, I had my computer guy (who was cheap coz he was on an annual maintenance contract) remove the hard drive from the old PC and put it into the new one, not only giving me my old data, but also 40 gb more of storage space.