If both of your computers are on a shared network, you can copy the music from the old computer to a folder on that shared network, and then download it onto your new computer. You'll want to delete the files from the shared folder once you've finished downloading so that your music doesn't take up space on the network, and so that others can't illegally download your music.
You can copy all of the music from your old computer onto an external hard drive or removable storage drive, often called a thumb drive or a jump drive. Once the music is all on the drive, just plug it into your new computer and copy it onto your hard drive or other desired location there.
Finally, you can store the music files to a personal account on a storage server. Once you've uploaded them from your old computer, you can download them onto your new computer. You can then either delete them from your account on the storage server or leave them there as backups.