Your first order of business after arriving on the island is reaching the top of the lighthouse, because that's where a transport will be waiting for you. Thankfully you won't encounter any resistance along the way, however you should be on a lookout for silver eagles (coins) and other less important objects. The game is similiar in this way to the previous BioShock titles, so you should explore all areas and examine all barrels, drawers, bags, wallets or even trashcans. The loot found in plain view is always the same, but the objects stored in containers are randomized. You can find the first set of silver coins in the roofed area of the pier and inside a barrel found near the lighthouse (to the left of the entrance).
Once you've entered the lighthouse make sure to explore each floor very thoroughly. The ground floor contains several coins which are stored in the dresser under the stairs. Once you're on the first floor search the locker and pick up a wallet found under the bed. The second floor of the lighthouse has several more coins and they can be found on a table next to a dead body of the (alleged) lighthouse keeper.
The top floor of the lighthouse has a door you need to open. You must ring three bells, following the combination displayed on the card - interact with the left bell once, with the middle bell twice and with the right bell twice. A red chair will appear behind the door in a few minutes. Sit on the chair to begin your journey to the city in the skies.