There appears to be at least 2 story arcs. You have to finish an islands tasks even at Home Island in order for the next island to become an available destination.
- Home island - starting island in game
- Nearby Island - require lvl 7
- Torch island - require lvl 7
- Coral island - require lvl 8
- Serendipity Island - require lvl 10
- Treasure island - require lvl 10
- Divination Island - available at or before lvl 23
- Sun island - require lvl 12
- Hippo island - require lvl 13
- Raccoon Island - available at or before lvl 23
- Lilac Island - require lvl 14
- Ribbiting island - require lvl 15
- Bouncy Island - require lvl 17
- Red island - require lvl 20
- Forest island - require lvl 23
- Lost island - require lvl 27
- Island of Tears - require lvl 31
- Storm island - require lvl 33
- Mysterious island - require lvl 35
- Pearl island - require lvl 36
- Muskelon island - require lvl 37
- Feral island - require lvl 38
- Gem island - require lvl 39
- Sapphire island - require lvl 40
- Invention island - require lvl 43
- Iron island - require lvl 45
- Frost island - require lvl 46
- Map island - require lvl 47
- Roaring island - require lvl 48
- Turtle island - require level 33
- Maze Island - require level 51
- Balancer Island - available at or before lvl 23