"Start" (and "beginning,") and "finish" (and "end,") are ordinary basic vocabulary words and we use them all the time.
"Onset" is a more literary word, and to me it suggests both the instant of starting and the events that unfold immediately after starting. "The boat race started at exactly 2 p.m. At the onset, all of the boats seemed evenly matched, but as time went on it became clear which were the fastest." It can also be used for something where the moment of beginning isn't quite definite. "My throat is tickling and I think this might be the onset of a cold."
"Coda" is quite unusual. It comes from a root that literally means "tail." I don't think it's used anywhere except as a technical term in classical music, for the end of a symphony.