2 Answers

Contact iTunes and tell them you got a new computer and they will send you download links for your new computer. 

There is a link called "remote assistance' which allows you to link another computer to your computer  and also another facility called "file sharing" . The former is for a computer connected via the net, the latter is within the same computer but also accessible through "remote assistance".

The best and most tedious method is to copy across the tunes via usb and its manually done. This way, the latter, you can also delete the stuff that you really do not need anymore, and have outgrown in taste. The procedure is caused because each of the computers is stand alone and not networked. You can also transfer files using the "attach large files" feature and send them across by email. Tedious I know, but at least you get them across.