When the music is downloaded, it goes to the Shared Music Playlists, but when I try to click and drag the music to my library or my iPod, it won't follow. How do I fix this problem?

Do you mean within iTunes? That 'LimeWire Tunes' is just a playlist - so it doesn't contain any actual music files. That's why you can't move files from there. That playlist hasn't been created within iTunes - it's just a pointer to where your LW files are on your HDD. So that iTunes can play the files.

Try moving files from LW's saved folder to iTunes. Some people can drag & drop files directly from LW's library into iTunes, other people find that this just doesn't work for them. Check the thread below for lots of info & different ways to add files to iTunes
How to transfer music to my iPod (& iTunes)

Thanks anyway, but like, right before you answered my question I figured it out.