The Swiss Family Robinson faced a multitude of problems on the island after being abandoned by the crew in the storm. The novel reads practically like a handbook on problem-solving.
Problem #1: Getting the Family to Safety.
The crew took all of the lifeboats, leaving the family to die on the ship when it breaks up on the rocks. Fortunately, the father and sons devise a clever homemade raft of barrels and life jackets to make reach the island safely.
Problem #2: Dangerous Wildlife
Poisonous snakes, sharks, 30 foot long boa constrictors, packs of jackals, porcupines, bears. The island is full of dangerous creatures.
Problem #3: Surviving the Elements
The family has to carve out their own existence on the island, learning how to live off the land and beat their thirst and hunger, as well as protect themselves from the heat and rainy seasons.
There are several problems the Robinson family encounters over the course of the novel.
Problem 1: Surviving the shipwreck. The other passengers and crew have taken the lifeboats and the ship has capsized on a reef. Thus the first task they need to accomplish is to get from the capsized ship to the island they can see. To solve this they build a raft and use it to not only get to the island but to carry with them all the useful materials they can scavenge.
Problem 2: Protection from Elements. In order to survive, they need to be protected from the weather, including rain and cold, and wild animals. At first, they build a tent and gradually improve it, but eventually they discover a cave that becomes a more secure home.
Problem 3: Food. They need to discover reliable sources of food to avoid starving. This initially means hunting and gathering, but over the long term it means agriculture to provide a more secure and reliable food supply. They also need to create secure ways of storing food safe from monkeys and other wildlife.