How do I transfer my library from an old computer to a new one?

If you want to do it via Home Sharing, playlist information won't come across, but we should be able to get the content to be transferred across.

First, make sure you've got the home share set up on both computers, as per the following document:

iTunes: Setting up Home Sharing on your computer

Have both computers running and have iTunes open on both computers.

Now, in the iTunes on the computer you want to transfer music to, select the library that you want to transfer music from, as per the following screenshot:

Once the other library loads, go down to the bottom of the screen and in "Show:" select "Items not in my library":

If you want to import all of the songs you can now see, head up to the iTunes menu bar and go "Edit > Select all". Then head down to the bottom of the screen again and click the "Import" button in the bottom-right-hand corner.