A USB flash drive would be the easiest way to transfer files. You can get them online or at just about any electronics or office supply store, or even Target. Here’s an example of an inexpensive one from Office Depot:
You would simply plug it in to your current PC and it will show up as a new drive. Copy your files to it, then plug it into the new machine and move the files to their new home.
16 gigabyte models like the one I linked to are pretty cheap, and even 32 gigabyte ones are easy to find for under $30.
Even if you copy your files to the new computer, I would recommend removing the hard drive from the old system and keeping it. If you ever discover that you’ve missed an important file, it would be easy to reconnect the old drive to your new computer to retrieve it. Plus you don’t have to worry about erasing the drive before you get rid of it, although that is pretty easy to do as well.
Feel free to drop me an email if you have any questions as I’d be more than happy to help!