Proceed means To continue as planned. Go on means To continue or move to the next thing. Carry on means To continue doing something or to cause something to continue. Continue means To continue keep on means To continue to do something or do something again and again. I think, these all have almost the same meaning. ...

Proceed = to go forward with something Example: Someone explains an order to you, and you have approved everything. Then you would tell them "please proceed with the order". Go on = to continue OR to go ahead with something Examples: "The show must go on." You would say this if something happened to make it difficult for the show to still happen, but it must happen anyways. Carry on = to continue what you were doing (if you were interrupted and had to stop doing something) OR if someone is talking a lot about a specific subject, that person is said to be carrying on about that subject. "She really carries on about her daughter" Continue = to keep doing something Examples: He continues to go to the gym everyday. Continue down the road for 3 miles, then turn right. After he broke his leg, he could not continue to play football until it healed. (he could not play any longer until it was healed). You could say "they continued with the show" or "the show went on". These have the same meaning here. I would say "go on" is probably more colloquial. Other than that, I cannot think of an example where any other words would be interchangeable. They have different meanings, although sometimes subtle. I hope all of this makes sense! :) If you have any questions or need more explanation, just let me know. :)