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 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 the new computer, you can add them its iTunes library. However, you are likely to lose supporting data, such as playlists, ratings, playcount, etc. Song data embedded in the song file will be retained, such as artist, album, genre, etc. There many be specific transfer utilities that support transfer of playlists.