When you go ashore you have to bring your room card. If you are going to the private island probably don;t need your passport. Some of the ports want picture id for when you get back on, I used a county id card not my passport. Or a picture license is fine.
Usually, the room card is your money, everything gets charged to that so I wouldn;t worry about carrying any money or leaving stuff on the beach. You can purchase a plastic waterproof little pouch that hangs around your neck, a couple of dollars and keep you room key in that, which by the way is encrypted with your picture. The tenders run back and forth all day usually every 15 or 20 minutes, so you are not captive there if you want to go back to the ship. We missed the island on our trip, bad weather. I wrote something about it but apparantly no one was interested enough to read it. You can sign up for tours at the excursion desk or do it online and the tickets are sitting in your room when you arrive. Happy Cruising.