Finny definitely has a personal moral code, and the best example of it is when he breaks the school record for swimming, but refuses to tell anyone about it. In fact, he outright tells Gene not to mention it to anyone. So, he just accomplished this major feat, and refuses to brag. This suggests that he enjoys challenging himself, and setting new heights for himself, but that the act is only for his own satisfaction. He doesn't want to bring others down or make them feel bad just because he's good at something. This is also seen in all of the sports awards that he has won--he has one numerous awards, all related to being a good sport, being a good competitor, and setting an example of comraderie on the teams. So, he plays and does things for the challenge, not to put other people down. His moral codes states challenge yourself, but not by stepping on others. The way that he includes everyone in games also states that he believes in making people feel good about themselves. He also believes that you should be doing all that you can in any situation, as seen in his desire to be a part of the action in the war. He doesn't have a problem breaking established rules at the school, as long as breaking them fits his other codes of challenging himself, including others, having fun, and being a part of the action.
Gene's moral code follows along with more standardized rules; he doesn't like breaking rules and is very uncomfortable with it. He even gets mad when Finny gets away with it--he feels people should be punished for breaking rules. He doesn't like lying--when Finny says he's his best friend, Gene doesn't lie and reciprocate the compliment, because he knows he doesn't feel that way. So, he's rather truthful. It bugs him that Finny doesn't know the truth about the tree, and when Brinker confronts him in the boiler room, Gene gets horrifically uncomfortable when faced with the prospect of lying. So, Gene is a more "by-the-books" guy when it comes to honesty and following rules.
I hope that those examples helped; good luck!