First of all, most of the initial islands have no gold. Open a game save in Don't Starve Beach mode.
So first make a raft.
Then cross the sea to the second island.
How to find the island?
The easiest way is to look at the color of the sea water. There are islands only in shallow seas. The eye of this sea is where the deep sea and shallow sea meet. You can find islands by going in the direction of the shallow sea.
The deep sea is where there are waves. You can find islands by crossing the deep sea and reaching the shallow sea.
