"synching" IPOD mini to new pc

I'm not sure from your post whether or not your iPod is appearing in iTunes or not. However if it is being picked up by your computer and 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
iPodRip Mac & Windows
YamiPod Mac and Windows
Music Rescue 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 - Windows version only)

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

When you get the 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". Your iPod should appear in the iTunes source list and you can change the settings to enable disk use.

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 7.3 or later 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. When your iPod appears in the iTunes source list you can change the settings: iTunes - Keyboard Shortcuts for Windows