I think 'proceeded' is more natural in the first sentence than 'went on'.
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Gene93Jack put his hand up and the bartender proceeded to pour him a drink.It sounds all right to me, but you can just as easily write ... and the bartender poured him a drink.
Gene93What is wrong with "go on", Jim?Nothing except what I already said, and that is a completely subjective feeling. I have no objective evidence that "go on" is wrong in the context you presented.
Gene93 I think that "doing something after doing something else" is confusing me. The bartender proceeds to pour him a drink, but that's the only thing he does.The man puts his hand up—that is the preceding action.
Gene93Does that mean that one person could do something and then another one could proceed to do whatever is necessary?Yes—or unnecessary. The two actions need have no logical relationship.
Gene93. Can "go ahead" be used as a synonym as it probably sounds more natural than go on?5jj has already commented on 'go on'. 'Go ahead' carries a different meaning—to continue in a process against some complaint, restriction, or warning. So 'go ahead' will not work in that context.