Talk to them. If you really want your friendship to be better again, just talk. That's the thing with true friends, if you talk to them and they listen, that's pretty much all the friendship you'll need. If all else fails, and it seems to keep getting worse, find another friend. Or connect with some former friends to talk to, because nothing's worse than trying to talk to someone who doesn't want to listen.
Well, if your a person of jokes, and your friends a person of seriousness, that's not a good friendship. Me and my friends are always silly with each other, but the serious people just don't get me. My advice would be to not apologize, not beg for forgiveness, but just smile when you see eachother, say "Hi" sometimes, etc. but don't be silly around that friend anymore?
The best way to make up with your friend is to apologize to them for what you did wrong. Make sure that you are sincere with your apologies. If your friend is still angry with you give them some time to calm down.