"Nano shuffle"? It's either a nano or a shuffle, but not both...
In any case (same answer either way), you cannot transfer songs from iPod to computer, using iTunes, except for songs you purchased from the iTunes Store (and you can also re-download purchases from the iTunes Store Purchased page). However, there are third-party methods and utilities that can be used to transfer song files from iPod to computer. If you do a Google search on something like "ipod transfer," you should get some links.
Once the song files are on your computer, you can add them to the iTunes library by dragging the song files (or the folder containing the files) to the iTunes window. Once the songs are available in your iTunes library, you can sync your iPod to your new PC.