How to make sure that all songs sync to the iPod

First, remove all songs from the device (from within iTunes) and then try re-adding them. If that fails to have any effect, remove all songs, unplug the iPod, re-plug, and add a few back. (Don't bother checking for songs in Explorer; that's not going to help.)

If that doesn't work, back up any data that has made it to your iPod and restore the firmware in iTunes. Then try to add a single song and see if that works. (Add a playlist with one song, then try dragging one over manually, if you prefer that setting.)

Comment on what worked or didn't: if nothing works, I'll come up with more things to try.

Your question can be rewritten in short form:

1. iTunes: How to move your music to a new computer
2. Where are my iOS backups stored? - expand the section About Syncing and Backups and you'll see

iTunes places the backup files in the following places:

• Mac: ~/Library/Application Support/MobileSync/Backup/
• Windows XP: \Documents and Settings(username)\Application Data\Apple Computer\MobileSync\Backup\
• Windows Vista and Windows 7: \Users(username)\AppData\Roaming\Apple Computer\MobileSync\Backup\

I linked to the official articles since things do change and cutting and pasting the steps today might not be right if things change in a month or two. Don't worry - the article about music moves everything except the iOS device backups. If your device is working, there's no need to move backups unless you want to keep older ones or save time the first time you sync with the new iTunes.