The Hero's Path

Personally, I first cleared the entire cave, explored all the islands, and then, seeing the color of the teleporters on each island, visited them all within a week, without using beacons or movement potions. I think it's enough to jump into the circles of the correct color; after a few iterations, you will definitely be thrown into the right portal. I didn't think there was any logic to it; I assumed that you simply end up in a random one of the same color (but not from those on the same island, as there are islands with identical portals).