Hey, Ikura. (My grandparents were from Norway and I have lots of relatives there; really need to visit some time)! Anyway, if I were you with one week to spend, I would stay entirely in South Florida. You mentioned shopping and dining; if you are arriving at PBI you should most certainly shop Wirth Avenue in Palm Beach. After that, I'd head down to the Miami area to see South Beach and have dinner at Joe's Stone Crab--an iconic restaurant with good food and a lot of history. Finally, save a couple of days to visit the Keys; drive all the way down to Key West if you can (3-4 hours from Miami). The scenery along the Overseas Highway is spectacular and Key West itself is a lot of fun and very laid back. You can rent bikes and explore the island, or just walk around.
Certainly, there is much to see and do on the West Coast and the Panhandle as well, but you're looking at several hours of driving to get there. You will have more time to spend having fun if you stay in South Florida and the Keys, then leave from FLL (or MIA, unless that's not an option).
Have a great trip, and next time come and see all the Norweigans in Minnesota!
Bill.