Probably the most simple procedure I can think of would be to put whatever files you want to move onto a flash drive, and then just move that between computers and pull all the files onto your new hard drive. Depending on how much you want to move, and the size of your flash drive(s)/memory stick(s), this might take a while, but it's almost certainly the fastest way to move things without physically connecting the computers.
A second option, which I think will depend on which operating systems the computers are running, would be a direct file share through the local network (that is, the network created by every device in your house connected to a router). This is something I have not done in a long time, and honestly, the files will probably be slower to move, and this can be frustrating to setup if you aren't computer savvy.
The third big option, if you are somewhat comfortable with the workings of a computer, would be to plug the old hard drive directly into the new computer. If you want to do this as a permanent second hard drive (and why not, if its still good), then it needs to be mounted in the case. Otherwise, you can unplug it from the old computers motherboard and power supply, and open up both computers, and just plug it into the new one. The computer will be fine to run with the side off as long as you aren't gaming on it.
The motherboard will label it's ports for plugging things into it. I assume your hard drive uses a SATA cable, but when you unplug it, just read the name of whatever it was plugged into and find the same thing on the new computers motherboard. There will probably be a few open spots, any of them are fine. The power supply is simpler, since those cables are brightly coloured and wont let you plug things into the wrong spot. You can also find tutorials for installing hard drives on youtube you can follow along with. Computers look really complicated on the inside, but having done this myself, it's not too bad.
If you go with the third option, be aware that electronics react very poorly to static electricity (the high voltage of a shock will fry the transistors), so you do have to ground yourself and the computer when handling the internal components. The simplest way on most computers is to leave them plugged in and flip the rocker switch on the back (the plug grounds the computer, the switch kills all the power), hit the power button to discharge all the capacitors inside the computer, and then ground yourself by touching any bare metal part of the computer case/chassis. As long as you touch it, you are grounded. I would also advise wearing cotton (which wont build up a charge against your skin), and working on wood (avoid polyester and carpet like the plague) and/or plastic surfaces (accidents can't happen if your environment can't cause a charge to build up).
