First of all, I should emphasize this important point:
Volcano Island is a turn-based strategy game. It has rather complicated rules and takes some time to get into. But if you give the game a chance, be sure that you'll love it and play it for months, or even years.
The goal is to rescue all the islanders
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To rescue the islanders, you should move the people to nearby ships. In Level 3, you can use also use choppers.
Challenges
There are two types of challenges:
blood thirsty creatures and
people who can't swim. There's only one way to cope with creatures: keep away from them, there's no tool to destroy them neither to make them go away. They'll be around you all the time.

You can use lifeboats to protect swimmers from
sharks, but they're not effective against
octopuses.
Many of the islanders cannot swim. You should provide someone who can swim on the same hexagon. Keep in mind that each hexagon can contain maximum two people.
Moving People

Moving people and boats are easy. Just drag and drop them. When you start dragging a person or a boat, a
magnifier appears. On the magnifier. you can see the person or the boat you're dragging. Be sure that you are moving the right person. If there are two people on the same place, pressing once selects the first person. Pressing twice selects the other one.
Also on the magnifier, you can see the
cost of the move. You generally have 3 moves on each turn, but this may increase if you have
bonus moves. Of course, on the first turn of each level, you have more moves.
Eventually, you can move the person within your limits. If the cost of the move is above your limit on this turn, the number on the magnifier turns red, which means that you cannot make this move.
Making the optimum move within your moves limit is the essential of Volcano Island. 
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Gaining Points

This is rather complicated. In normal conditions, you gain 1 point for each person you save. But there's more. In any turn, when you save a person, the next move becomes a "
x2" opportunity. Which means, the next move, the persons will be multiplied by 2.
And in this move, if you succeed to save anyone, the next move now becomes a "
x3" opportunity. The points will be tripled. And any consequent moves with saved people will continue this "
x3" opportunity.
If you fail to save someone in these "
x2" or "
x3" opportunity states, then the spell is broken and you return to "
x1" mode again.
It looks complicated, but you can get it more clearly when you play and observe the way your points grow. On the scoreboard, you can see all the necessary information you need. Also you can see the actual
point multiplier. If you calculate your moves according to the point multiplier, you increase your high scores enormously.
Lives / Tolerable Losses

Of course, the ultimate goal is to rescue all the people. But you have always some margins. For example, in the first level, you have 2 tolerable losses, in the second level you have 1 and in the third level, unfortunately none. But you can overcome this hardship by carrying your
unused lives to next levels.
For example, you have 2 lives in the first level. If you complete the first level with no loss or one loss at most, you carry a bonus life to the second level. On the other hand, if you consume 2 lives, you pass to the second level but you can't carry a bonus life.
Continuing with the example, you carried this bonus life to the second level. You have the luxury to spend this bonus life. If you spend this one bonus life at most, you again carry one bonus life to the third level. Again, the best way is to play and see how it works.