Can I move goods between islands automatically?

Yes, it's possible to move goods between your islands automatically. You just have to set up a trade route. You'll want to have a ship that you can assign to the route, so if you've only got one I'd suggest building or buying a second, just so that you can still manually move goods around (ie, for founding new settlements on new islands and such)

1. Click the "Strategic Map" button in the lower right (it looks like two arrows going in a circle) or press F6.
2. Click "New Trading Route" in the upper right.
3. (Optional) Double click on the new route's name to change it
4. Click on the map icon for the "source" island - where the goods are created that you want to move. An entry will be created in the "Stations" list for this island.
5. Click the green button under the station name, and select the good you wish to load on the ship.
6. Repeat the previous step until all goods are loaded (bearing in mind the carrying capacity of the ship you're planning to use)
7. Click on the map icon for the island where you want to drop the goods off
8. Click the red button under the station name, and select the same goods that you did in the step with the source island earlier
9. Below the "Stations" section, you'll see the "Ships" section - click "Add Ship" and then pick the ship you want assigned to the route

The ship will head to the source island, pick up the goods, and deliver them to the destination. It's possible to make more complex chains - pick up on 2 islands and deliver to one, for example, or drop off part of some goods on one island and the rest on another.

Here's a video tutorial to the process: