I have now successfully downloaded some tunes, they appear my iTunes library but transfer to a Shared file called "Sharon's Limewire Tunes". When I sync they do not transfer to my iPod device. How do I move them to my device? I have tried to drag but nothing happens?

'Sharon's Limewire Tunes' is just a playlist. Those files aren't in the iTunes library, that's why you can't transfer them to your iPod. You'll need to add files from LW's saved folder to the iTunes library.

In LW, tools > options > saving will show you the location of your saved folder. If you go to LW's library, highlight a song & click Explore...your saved folder will open. Add files to the iTunes library from this folder

There's lots of info here.
Transferring music to iPod & iTunes

Thank you for that Birdy and I have now transferred it to my iPod successfully, it seems so easy now I have done it.

How do I add music and DVD movies from my collection to limewire so other people can use them?