Anyone got a list of things what you can bring for the island later on in the game and items you have to leave behind?
You can't take anything to the island except what you're wearing. No tools or items of any sort except clothing items that you are currently wearing. Note that wetsuits you are currently wearing count as clothes (so you can take them) but can still be used as tools on the island. Though this isn't a huge advantage as you can rent a wetsuit on the island anyway.
Tools instead will be provided by a Gyroid at the island and on each of the tours. You can take any non-tour related item back to your town by leaving it in the basket in the hut when you first reach the island. All other items will be sold at an extremely low price when you go home or go on a tour.