Install Itunes on new computer.
On your old computer, in Itunes, click File > Library > Export library
Transfer file that is created to the other computer.
Transfer the music to the other computer, make sure the file path remains the same.
On your new computer, do something with the created file. I think you have to drag it to Itunes or something.
If all goes well, your music should all now show up in Itunes on the new computer.
Good luck!