Once you have the new computer, install the iTunes software on it - just download from the iTunes website again.
Then, use a USB cable from the old computer to the new one to add the iTunes files as follows: click on top-left corner of iTunes on the new computer - I think it says “file” - and find “add folder to iTunes.” When you click on itm, find the file which on a Windows computer would be marked “My Documents/My Music/iTunes (again, the file on the old computer). Click on that, and the program should copy all the tunes onto your new hard drive. (Or, you could just downloady the My Music/iTunes file to a storage drive and plug it in to the new computer - same thing.)
Just connecting your old iPod to your new computer will NOT accomplish this - it might just erase everything from the iPod. In fact, once you have all the songs on the new computer, then you will have to “re-format” your iPod, since the new computer will not recognize it. The computer will re-add the songs back on the iPod.
There may be a way to choose “manage music manually” option/preference (on one of the drop-down menus at top left), and just load everything directly from the iPod, but I never did it that way.
Email me if you want me to try to talk you through. Or you might get better advice from others. But I have done this kind of thing myself, successfully. Good luck.
Yes, you can request “manually manage songs and videos” when you plug the iPod in for the first time.
Older iPods cannot be used to transfer anything but iTunes purchased music to a computer, alas. I don’t know if the newer ones have data transfer capability.