You cannot use iTunes to transfer song files from iPod to computer (except for songs purchased from the iTunes Store). However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links. (The CNET articls seems to be popular.)
Once the song files are on your new computer's drive, you can add them to your new iTunes library.
NOTE: Songs that you purchased from the iTunes Store can be downloaded again at no cost. In the iTunes Store, sign in using the same Apple ID used to make previous purchases. From the iTunes Store Home screen, click Purchased under QUICK LINKS. On the Purchased screen, you can find and download previous purchases.