How can I use a new computer without deleting everything on iPod?

1). 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.

2). When your iPod appears in the iTunes source list change the update setting to manual, that will let you use your iPod without the risk of accidentally erasing it. You can continue to use your iPod and add to it by dragging and dropping content from iTunes: Managing content manually on iPod and iPhone

If you wish you can stop there and just keep using your iPod in manual mode. However, once you are safely connected there are a few things you can do to restore your iTunes from the iPod.

3). If you only want to copy your purchases to the computer that can be done using iTunes, you'll find details in this article: Copying iTunes Store purchases from your iPod or iPhone to a computer

For everything else (including purchases) 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:
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

4). 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.

*Ok, I don't want to move everything on my other computer's iTunes to my new computer. I just want to be able to use my iPod on the new computer.*

So you can stop at the section where I've said "if you wish you can stop there and just keep using your iPod in manual mode."

*So, just so I'm sure, I can plug in my iPod, without worrying,*

If you take care and choose Cancel at the point I've indicated and if you change the option from syncing music to managing the iPod manually then after that you'll be able to connect and you won't get the message again.

There is an alternative method to stopping an iPod syncing if you want to know it. You can also use a keyboard command to prevent your iPod auto-syncing with iTunes. While connecting the iPod to the computer on Windows with iTunes installed hold down the Shift + Ctrl keys together. This will stop the iPod from auto-syncing with iTunes and the iPod will appear in the source list. Wait until you are sure the iPod has mounted, and that it will not auto sync and then you can let the keys go. This may take between 20 to 30 seconds depending on your computer. Once you are connected you can change to manual update as in the original post: iTunes - Keyboard Shortcuts for Windows

In either event, which ever method you choose you just need to take care. If you don't feel confident doing it then I'm afraid there's nothing more I can suggest.