No you won't have a problem installing iTunes on another computer. I can't recall there being any problem with serial numbers with iTunes and you can even freely download the latest versions from
http://www.itunes.com The iPod is just great. I bought my wife an iPod Photo with a 40GB drive a year back and she has loaded probably 75% of our music collection onto it. I also got her the FM transmitter that plugs onto the top of it and now whenever we go out in the car we just take the iPod and tune the car radio into it. Plus recently I was in a hotel before a job on New Years Eve and wanted to run over a new song we were playing. I turned on the iPod and tuned the clock radio into it so I could play along with the song I'd put onto my wife's iPod.
They really are a wonderful thing.
[This message has been edited by Nigel (edited 02-02-2006).]