How much music roughly in Gigabytes do you have to transfer?
A simple option for DRM-free music is to burn the music to several DVDs on the source PC and transfer the contents of these discs to the new PC. This way you also have a hard copy on discs. Alternatively, use a large (1GB+) pen drive to copy a certain set at a time.
If you are familiar with home networking and can connect the two PCs either using a network switch, router or cross-over network cable, you can network the two if you are familiar with home networking. This is probably the handiest way to transfer a large quantity of music and other content (e.g. 30GB+), as you can leave the PCs unattended copying without swapping discs, etc.
For DRM (copy protected) music, I would recommend checking the store’s FAQ or help section to see if there is any special process to transferring the music, such as how to deactivate the original PC and activate the new PC for the DRM. Another option is to burn the music to CDs on the original PC and re-rip them on the new PC, which effectively strips the music of its DRM also. This may be the only option if you’ve reached the limit of devices you can play your DRM protected music on.