Go ahead - can be used in different situations. For example, say you were in a heated argument and a person stops mid way through an insult you can reply "Go ahead, say what you were going to say." But go ahead can also be used in a different context for example, if two people are trying to walk through the same door one person could say "go ahead, you go first" Carry on- this means to keep doing what you're doing. For example, if your boss were to interrupt you while you are working, when they have finished talking to you they might say "carry on" I've personally never heard anyone use the phrase keep on. But it might mean keep on doing what you are doing.

Eg Go ahead! Try it! He carried on (continued) talking to his friend when the manager told him to concentrate on his work