I've avoided the obvious "don't message your ex" rule and focused on some that may not be so obvious to some people. They're not too hard to follow, and they could probably save you some arguments.
1) Both change your FB status to "in a relationship". It may not be necessary to tag the person, or share the event on your wall, but just make it clear for those who may still be curious. And guys, no need to wait for your gf to ask for it.
2) Post pictures of you with your s/o...on all platforms. I see a lot of guys who have pics with their gf on FB because she tagged them in them, yet their Instagram is filled with topless selfies and no gf in sight...Just do it here and there, no need to be having couple selfie photo shoots every night either.
3) Do not accept friend requests of the opposite sex if you do not know who they are. Please, we're not 14 anymore and FB was not invented yesterday, and you don't need 500 guys or girls that you don't even know.
4) Don't just talk to a "friend" of the opposite sex everyday on social media...If your s/o knows them and knows exactly how platonic your relationship is, it may be fine. If however this is just a random friend that you don't even see anymore, and you're just talking about random stuff just to talk, don't, just don't.
5) Don't change your profile pic when you're in a fight. Really I mean what is that going to achieve in resolving your argument?
6) Don't write nice things to or about your s/o after a fight is over. We're living enough through social media. Just tell them in person. It's way better for you, and way better for the rest of us who just don't care.
7) I don't think its really necessary to reply to random's (of the opposite sex) comments on Instagram...
8) If you're fighting, don't post quotes or lame pics to let the rest of us know you're having relationship problems. This is not what social media was made for. It's annoying. Please just read those to yourself if they make you feel better.
9) Don't have/follow any of your ex's on social media. You're probably not friends anymore. So why the f do you need to be FB friends?!?!
10) If you're fighting, and you're not together, and you're MIA (not answering calls or texts)...don't be active on social media, it's just f***ing aggravating. Call. Text. Go home and figure it out. Social media will always be there. Social media can wait.
* Please note that in any case "of the opposite sex" can also be replaced by "of the same sex", depending on your sexuality.