If you are using automatic syncing to load the iPod, all songs on the iPod are in your iTunes music library. So, you won't need to transfer anything from the old nano. Just sync the new iPod from your iTunes library the same way.
If you use the Manually manage music setting, AND there are songs on the iPod that are not in your iTunes music library, you will need to off-load songs to your computer first. You cannot transfer song files from iPod to computer using iTunes (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 a Google search on something like "ipod transfer" you should get some links.
Also, if this is a 1st gen iPod nano, you can check this article to see if it applies to you