One thing they have to overcome is the potential for chaos and irresponsibility. In order to get the things that they need (food, shelter, a way to attract passing ships), they must work together. So, initially, they need order. In the first chapter, Ralph and Piggy find the conch. They use this to summon the boys. When a meeting is organized, only the boy holding the conch may speak. This creates an order and organization. Ralph delegates jobs to the boys. They will build shelters, take turns watching the fire, and some will hunt.
The major technological hurdle they overcome is how to build a fire. They consider rubbing sticks together and Roger suggests using a bow. But then it is actually Jack who suggests using Piggy's glasses. With his glasses ("specs"), they are able to start the fire. As long as they keep the fire going, there is hope of attracting the attention of a ship. And, they will eventually be able to use the fire to cook meat.
But the largest obstacle that they must overcome is themselves. Ralph tries to sustain organization and order with Piggy and Simon. But, Jack becomes less and less civilized and lures most of the boys to his camp. In the end, Ralph is unable to prevent their descent from order to chaos, from civilized to savage. That said, they initially overcome this impulse but inevitably fail in sustaining it.