Normally, we just say 'I'm going fishing' to talk about fishing or 'I'm going hunting' to talk about hunting. If you say that you are going to hunt an animal or catch a fish, this implies that there is one specific animal/fish that you want to hunt/catch for a reason. Why do we not use hunt when talking about fishing is a good question. I'm afraid that I can't answer this for you because I didn't decide on the rules when the English language first started. Both words mean the same thing e.g you want to kill an animal/a fish. The English language has lots of ridiculous rules. The important thing is NEVER BE AFRAID OF MAKING MISTAKES. I hope that my explanation has helped you.

We catch fish because to hunt means one is following to obtain an object. With fishing, you are standing still, using a fishing g rod and waiting for the fish to bite. Of course, in some cultures they may hunt using a different tool to catch fish. It depends on the situation.