I just purchased an older, but new, ipod and I am totally ignorant about how to use it. It charges fine and I can download music from my computer into it and upload the music into another computer, but I can't get any sound from the headphones. Is some software or driver program required to activate the ipod?
No. You should just be able to plug your headphones in after syncing music and be good to go. Have you double check to make sure the headphones are plugged ALL the way in, meaning the white part of the headset if flush with the body of the iPod Shuffle.
Perhaps it wouldn't hurt to also try another set or earbuds if at all possible.