How do I switch my ipod to a new computer without losing all of my songs?

Connect your iPod to your computer. If it is set to update automatically you'll get a message that it is linked to a different library and asking if you want to link to this one and replace all your songs etc, press "Cancel". Pressing "Erase and Sync" will irretrievably remove all the songs from your iPod. When your iPod appears in the iTunes source list change the update setting to manual, that will let you use our iPod without the risk of accidentally erasing it. Check the "manually manage music and videos" box in Summary then press the Apply button. Also when using most of the utilities listed below your iPod needs to be enabled for disc use, changing to manual update will do this by default: Managing content manually on iPod and iPhone

Once you are safely connected there are a few things you can do to restore your iTunes from the iPod. If you have any iTunes Music Store purchases the transfer of purchased content from the iPod to authorised computers was introduced with iTunes 7. You'll find details in this article: Copying iTunes Store purchases from your iPod or iPhone to a computer

The transfer of content from sources such as songs imported from CD is designed by default to be one way from iTunes to iPod. However 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. Have a look at the web pages and documentation, this is just a small selection of what's available, they are generally quite straightforward. You can read reviews and comparisons of some of them here:
Wired News - Rescue Your Stranded Tunes
Comparison of iPod managers

TuneJack Windows Only (iPhone and iPod Touch compatible)
iPod2PC Windows Only
iDump Windows Only
SharePod Windows Only (iPhone and iPod Touch compatible)

iPodRip Mac & Windows
YamiPod Mac and Windows
Music Rescue Mac & Windows
iPod Music Liberator Mac & Windows
Floola Mac & Windows
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

Whichever of these retrieval methods you choose, keep your iPod in manual mode until you have reloaded your iTunes and you are happy with your playlists etc then it will be safe to return it auto-sync. I would also advise that you get yourself an external hard drive and back your stuff up, relying on an iPod as your sole backup is not a good idea and external drives are comparatively inexpensive these days, you can get loads of storage for a reasonable outlay.

Unfortunately you cannot take songs from your iPod and put them in your library (even thought they came from your library originally... on another computer). I had this same situation happen to me and have read through all of the literature and talked to a few Apple "professionals" and they relay to me the impossibility of this. They say it would be technically considered illegal file sharing. Even though these are your songs, if you were able to put them onto just anybody's library then they would then own a song that they did not purchase legally. It is unfortunate because you are NOT trying to commit any crimes, you just want your darn songs back in your library without having to rip them again. (Note: there are programs online that can help you perform this seemingly "heinous" act)

HOWEVER, any songs that you bought from the iTunes store that are on your iPod CAN be transferred to your new library. Connect your iPod, then Click on File menu, then "Transfer purchases from iPod". You may be prompted to authorize your computer which you should agree to. I hope you can take something from this.
Good luck.

I do want to add an additional note. You are not required to sync with your library. When you plug your iPod into the computer and it show up in iTunes, click on the device itself (not music or anything like that). On that screen (where it gives you the serial # and how much space it has left (blue, orange, purple, etc. bar)- there should be four options at the bottom. If you click manually add songs, you can keep all of the songs you have on your iPod. This just means that rather than clicking on the check boxes to add and delete songs from your iPod, you need to click and drag them onto your iPod or click on a song on your iPod and manually delete.
I think this one might've actually helped... maybe.
Peace.

Apple just came out with a legit Ipod copying program called MusicRescue, and its awesome! I just got my first Mac ever (MacBook) and needed to copy my Ipod Touch music to Itunes, update the Ipod Software to 2.1, and then reload my music to my Ipod. Its $16 for a license, but priceless for the ease and security. It did it all for me, I have 16Gigs of music, and the whole process took 3 hours, 1 hour for each move. So happy with the result. Good luck!

*Apple just came out with a legit Ipod copying program called MusicRescue.*

Music Rescue is not an Apple product, it's developed by KennettNet Software Limited and it's been available for quite a while now. It's one of many third party utilities that are linked to in the Apple software download pages. You'll also see that it's in the list of possible utilities I originally posted back in September when this question was first asked.