My grandmother's family is from Capri
I spend atleast a week there every year! Be sure to try the "Limoncello" in Sorrento (it's a very nice lemon-based alcoholic drink).I can tell you a bit about Capri!
What I recommend is the view of the sea from Monte Solaro, and bring some good shoes, as most tourists are not used to climbing the stairs (and the best way to get around is on foot).Also the Grotta Azzurra is a must see!We used to swim there when I was younger, although I think it's forbidden now. You are going to have so much fun
If you want to go out at night, the Underground is a nice place (it's in the center of Anacapri).The other big one is the Discoteca Zeus but I didn't really like that one
As for shopping, there's a Roberto Cavalli shop for sure, and there's a boutique called La Parisienne although I'm not sure what brands they sell (as usually I'm not shopping when I'm there). Maybe there's a Fendi (or
Gucci?) shop, but I'm not 100% sure on that.
Edited to add: Be VERY VERY careful in Napoli!Lots of pickpockets who look out for tourists to rob!Seriously, be really careful there!It's waay worse than in Rome.
If you have any questions I'll try to answer , my grandma still lives there to this day with my uncle's family!