The easiest way to do it would be to buy a USB transfer cable at (Wal-Mart, Target, RadioShack, Best Buy, other similar retailers). The cable allows you to transfer files from one computer to the other. This would work best if one or both computers is a laptop.

You can also transfer the music using a USB stick or flash drive. You just transfer the files to the USB’s memory from the first computer. Then, connect the USB drive to the second computer and transfer to the computer. USB sticks are fairly inexpensive. However, this way can take awhile, depending on how much music you have. They tend to have limited space (1-2GB).

The same method above can be used with an external hard-drive. They also have more space, so you could likely get all of your music on there to transfer and do it in one go. However, they are also significantly more expensive. While it can be useful to have one for file back-up, if you don’t foresee yourself using it for other purposes, it’s not worth the expense.

Finally, you can use CD-RWs to transfer the music. You would burn the music onto the disc then import it into iTunes on the second computer. However, you would likely need multiple discs and they can start to get fairly expensive.

Any of these methods can work to transfer the music to the new computer.

If you have the music on your old iPod, but not on the first computer, you will have to invest in a program to rip the music off of the iPod and onto the second computer; iTunes will only automatically transfer music purchased through the Music Store. A program that I’ve used and liked is Music Rescue. It costs about $20 and is very easy to use. You should only use this option if none of the other suggestions are viable for you.

Good luck