What you have on your computer is your iTunes library. The iTunes library is not directly tied to an Apple ID. The Apple ID is your iTunes Store account, used to make purchases. When you sign in using your Apple ID in iTunes, that let's you make purchases and download from the iTunes Store. It does not automatically change your iTunes library on your computer (unless you happen to be subscribed to Apple's iTunes Match service).
So, signing it to the iTunes Store using your Apple ID does not make things appear in your iTunes library. However, for songs (and other media) purchased from the iTunes Store in the past with that Apple ID, you can re-download them to your current iTunes library at no cost. This is only for purchased items, not songs that came from music CDs or other sources.
In iTunes, go to the iTunes Store Home screen. Sign in with your Apple ID, if you are not already. Click on Purchased under QUICK LINKS. This takes you to the Purchased screen, where you can find and download previous purchases.
If you have songs on your iPod that are no longer on a computer that you have access to, you can offload the song files to your current computer. You cannot use iTunes to transfer song files from iPod to computer, except for songs purchased from the iTunes Store. However, there are third-party methods and utilities that can transfer from iPod to computer. If you do an Internet search on something like "ipod transfer," you should get some links.
Once the song files are on your computer's drive, you can add them to your iTunes library. Then, you can sync your iPhone and iPod as desired.
