The word "coast" is used for oceans and large seas.  "Shore" can be used for oceans, seas, and lakes.  But for rivers, the best choice is "bank".

On the left bank of a river there is cabbage, a wolf, and a goat. The man has a boat, and he must deliver the cabbage, wolf, and goat to the right bank. The man can't leave the wolf and goat alone together, nor the cabbage and goat alone together. You understand why. The goat and wolf are hungry. The cabbage cant be hungry:)). The man in boat can carry only one object in the boat. How can he handle it?

Yesterday I asked children this question. They could not give me an answer. But was very lively/animated/spirited discussion. It was very interesting to listen to.

Here is my guess how the man can do it:

The man takes the goat to the right bank. This leaves the wolf and cabbage on the left bank.
The man goes back to the left bank. He then takes the cabbage to the right bank.
But when the man arrives at the right bank with the cabbage, he doesn’t leave the cabbage with the goat.  Instead, the man takes the goat back to the left bank.
The man leaves the goat on the left bank, and takes the wolf to the right bank, leaving it with the cabbage.
The man goes back to the left bank, and then takes the goat to the right bank.