What do the reactions of both Cesario(Viola) and Sir Andrew towards fighting a brave rival reveal about their characters in Shakespeare's Twelfth Night?
Since Cesario is actually a woman, she has no training in sword fighting; therefore, her show of cowardice at fighting Sir Andrew when challenged merely shows she knows her own limits, she doesn't have the skills to be able to fight a man. In addition, as she states, it also shows she is a peace lover, as we see in her lines:
I am one that would rather go with sir priest than sir knight: I care not who knows so much of my mettle. (III.iv.254-56)
In other words, she is saying here that she cares more for religion than fighting and doesn't care who knows she is a coward.
Sir Andrew, on the other hand, is a knight, and should be well trained in swordsmanship; therefore, his own cowardice shows that he is a great idiot who is incapable of thinking for himself. We especially see evidence of Sir Andrew's incapability of thinking for himself when we learn in the second scene of Act 3 that it is really Fabian and Sir Toby who came up with the idea of Sir Andrew challenging Cesario. In this scene, Sir Andrew is about to leave the estate because, upon Sir Toby's encouragement, he is trying to court Olivia but sees her paying more attention to Cesario, a messenger, than to himself. Fabian argues that she is not paying Sir Andrew attention because he is not fighting for the rights of her attention. He argues that Sir Andrew should have "bangled the youth" before Olivia, which certainly would have drawn her attention (III.ii.19). When Sir Andrew protests that as a knight he must act with valor, it's Sir Toby who comes up with the scheme of challenging Cesario to a duel. But unlike a typical knight, Sir Andrew completely loses his nerve when Sir Toby falsely praises Cesario's swordsmanship. Sir Andrew's sudden show of cowardice at something he himself initiated at others' suggestions proves that Sir Andrew is not only a coward but an absolute idiot with no mind of his own.