Island Saver

For this achievement, you have to acquire all 20 of the golden nest eggs in the video below is a guide to all of Fantasy Island you unlock the achievement at the end of The Island but the timestamps I'll put are for each egg
19:18 (Egg 1)
25:17 (Egg 2)
26:36 (Egg 3)
35:49 (Egg 4)
43:31 (Egg 5)
59:09 (Egg 6)
1:08:37 (Egg 7)
1:23:34 (Egg 8)
1:23:47 (Egg 9)
1:27:50 (Egg 10)
1:29:07 (Egg 11)
1:32:39 (Egg 12)
1:41:56 (Egg 13)
2:00:41 (Egg 14)
2:11:32 (Egg 15)
2:12:06 (Egg 16)
2:20:40 (Egg 17)
2:42:37 (Egg 18) for this one you have to complete the smoothie contracts to get the golden egg.
2:45:29 (egg 19) for this egg complete Forest Island and the spider will lower it down for you.
2:46:06 (Egg 20)
Then after collecting all the eggs the goose will give you the beanstalk seed to plant.

Story Achievement

The Golden Goose will reward you with the Giant Beanstalk seed after finding all her golden eggs, simply plant it in the large plot and water it for the achievement to pop.

If you are having trouble finding all the eggs there is a guide here;