Everything you need to know is here How to transfer music to my iPod (& iTunes)
Whatever you do, don't get confused with the LW playlist that shows in iTunes. It is simply a playlist that you can play, but can't copy or burn. You need to Add the songs to iTunes main library in the methods described in the above link. Once the songs are in iTunes Library, you can copy to iPod or burn to CD/dvd, etc. I would also highly recommend you use Manual updating of your iPod. See about this in the iPod tutorial & mention of it at the bottom of the page in my link.