(From San Juan to NYC; kept humming "America" from West Side Story) Our third Princess cruise, first in Caribbean. My wife and I stayed two nights in Old San Juan pre cruise. Old city is walkable; unless you are venturing further afield no car needed or recommended - traffic and parking are tough. Free trolleys will take you around the city conveniently. We stayed at Hotel El Milano - room and price were good, but very noisy at night - between two nightclubs - so can't recommend on a weekend. Carnival embarks from Old San Juan, walkable from Sheraton Old San Juan, but Princess and others from Pan American Pier, a cab ride away. If you wish, you can buy liquor at duty free and carry on board as long as you conceal it inside your carry on before going through security. Embarkation went fairly smoothly.
We had a balcony stateroom on Baja deck aft; very convenient. Top decks can be quite noisy with music and Movies Under the Stars all day, so I'd recommend if you want some peace and privacy. I don't think the recent drydock did much for the vessel; some areas were actually quite worn. The Piazza is new, but as others have noted, there was a bad odor from some of the cafe selections. Didn't like putting the Captain's Circle and future cruise desks in private offices; hard to see when they're there/available. Preferred the layout on Sapphire Princess. Pluses: did enjoy Movies Under the Stars in the evening; also, the Cafe Caribe concept was good for dinner if one did not want to go to the main dining room. Food was good; particularly the theme nights. Also, the Crown Grill has a no-extra charge "pub lunch" available with a limited menu; it was quite good. Ice cream (soft serve) is available all day free of charge; on the Sapphire it was limited to an hour in the afternoon. Minuses: dinner service in the anytime dining rooms was quite slow if one sat with a group; quicker if you have your own table for two.
St. Thomas: we took a ship's tour to Mountain Top for $29 pp - cheapest Princess shore excursion I've seen. We enjoyed the view and then back to Charlotte Amalie for shopping. I did the sightseeing walking tour; not that much there. Some people took the boat over to St. John for more natural scenery and enjoyed that. Unfortunately the ship docks at the new cruise terminal; not much there, so you have to take a taxi into town (easily available).
Antigua: we rented a car (first time driving on the "wrong" side of the road - we survived!). There is not much to see in St. Johns, and it is somewhat dirty. Driving wasn't bad once you get out of the city proper; just keep left! We drove around the island and stopped at a beach with hardly anyone there. We also visited the Royal Naval Dockyard, which I would highly recommend if you are at all interested in history. Betty's Hope plantation can be skipped; not much there.
Bermuda: I rented a scooter and took the ferry from Dockyard to St. George's, rode from St. George's to Hamilton, and then took ferry back to Dockyard, to see as much as possible. You can take your scooter on the ferry for an additional token. Note: if you don't get right off the ship, there will be a HUGE line to purchase ferry tokens. Stopped at Crystal Caves, which were interesting although not that large. Great tour guide. The ship was a little delayed disembarking, so could have missed the ferry. I had not ridden a motorized cycle before; I survived, but unless you are a competent rider, I'd recommend taking the bus, which runs frequently and is less stressful. If you want to see everything, I recommend taking my route; the ferry from Dockyard to St. George. St. George is very small and you can cover quickly; then take bus to Hamilton. There is a beach en route if you want to stop over. There's also a fair amount of shopping in the Dockyard area.
Disembarkation in Brooklyn was a little slow, but luggage was ready when we got off; we had a car service pick us up. Overall, an enjoyable trip - wanted to visit Puerto Rico, USVI, and Bermuda, so a good itinerary.