If I'm reading this right, you want to know the difference between the expressions of "come on" and "go on" when someone is reacting to information. If that's the case it's regional. Most Canadians would say "Oh, come on!" when being reactive, whereas most Brits would say, "Aw, go on!"

They both have multiple meanings. I might have forgotten some. "Go on." on its own can mean "continue", especially if you have paused while talking. It is an encouragement. "Go on." on its own can mean "Do it!" or "Do what I told you to!" encouragement or command. "Come on." on its own means "Let's go" or "Hurry up." "Oh go on!" shows disbelief, similar to "You're joking, aren't you?" "Oh come on!" is persuasive. "Come on, I want you to do this. You want to really..." "You do go on." means "Please stop talking. You go on and on and on." "That was a come on." means "That was flirtatious." or "S/He fancies you"

Hi Peter, go on means that you are proceed something or you are going from point a to point b but in this moment you are in the middle. Come on is an expression that can incitate or involve in a positive way someone to so something ( come on ! You can do it) or to hurry up and be quick. It also express unbeliveness ( come on !it is not be possible , you are kidding)