We did this trip in June. The hotels will have all of the soap and shampoo that you need.
Must haves: sunscreen and sunglasses, Chargers and charging cords, a plug/outlet adapter (I didn't see US plugs in our hotel rooms), something waterproof to put your bags/souvenirs in during the boat ride on Capri, swim mask or goggles (for Capri), comfy shoes. The mask/goggles was the one thing I didn't bring that I wished I had.
None of the dinners are particularly dressy, so you don't need any special clothes. Pompeii is very dusty, so you might want to dress accordingly that day.
We didn't get caught in the rain, but the guides did have ponchos (so no need to pack rain gear).
If I think of anything else, I'll post again, but we packed really light. Shorts/skirts and tees, no pants or jackets necessary. You might want to pack a sweater or cardigan. Shoulders must be covered to go in the Basilica in Pisa.