You need to copy the songs from the old iPod back to iTunes on your computer and add them to the new one from there.
iTunes will only give you the option to copy your iTunes Store purchases directly from an iPod to the computer, you'll find details in this article:
Copying iTunes Store purchases from your iPod or iPhone to a computer
For everything else (music from CDs, other downloads etc) there are a number of third party utilities that you can use to retrieve the music files and playlists from your iPod. You'll find that they have varying degrees of functionality and some will transfer movies, videos, photos, podcasts and games as well. You can read reviews and comparisons of some of them here:
iTunes & iPod software
Wired News - Rescue Your Stranded Tunes
Comparison of iPod managers
A selection of iPod to iTunes utilities:
TuneJack Windows Only (iPhone and iPod Touch compatible)
SharePod Windows Only (iPhone and iPod Touch compatible)
iPod2PC Windows Only
iDump Windows Only
YamiPod Mac and Windows
iPod Music Liberator Mac & Windows
Floola Mac & Windows
iPodRip Mac & Windows (iPhone and iPod Touch compatible)
iPod Music Liberator Mac & Windows (iPhone and iPod Touch compatible)
Music Rescue Mac & Windows (iPhone and iPod Touch compatible)
iGadget Mac & Windows (iPhone and iPod Touch compatible)
iRepo Mac & Windows (iPhone and iPod Touch compatible)
iPod Access Mac & Windows (iPhone and iPod Touch compatible)
TouchCopy Mac & Windows (iPhone and iPod Touch compatible)
There's also a manual method of copying songs from your iPod to a Mac or PC. The procedure is a bit involved and won't recover playlists but if you're interested it's available on page 2 at this link:
Copying Content from your iPod to your Computer - The Definitive Guide
