The main issue we've had so far has been deciding who gets to use the consoles and when. When we moved in we bought three video games; Rock Band 2 that we could both play, Mirror's Edge for Henrir because it was on sale, and Final Fantasy X for me because I wanted to try an RPG with a turn based combat system. It was easy to say "Hey, let's play Rock Band together," but hard to decide who got to play their individual game first.
Try as hard as you want, it is really hard for two people to play their individual game on the same tv. You will always want to play at the same time as your partner. One solution to this problem is to learn how to be a good video game watcher. When Henrir played Mirror's Edge I would sit on the couch with him and watch the game and give constructive help when necessary. Although I have to admit that it was a hard game for me to watch because I do not do well with first person and fast moving games. It's one of the very few things that give me motion sickness.
Since I started playing Final Fantasy X Henrir has watched my entire progress. He enjoys watching me play it so much I'm not allowed to play when he's at work so he can see what happens, even though he's beat the game before. I'll ask his opinion on how to fight some bosses and we discuss strategy and the usefulness of spells. I even let him do the sphere grid for some characters. Although I'm playing the game by myself it's been nice to have him next to me and being able to talk about what's going on with him. This makes playing a game less isolating and more enjoyable for me.
It's important to remember that when watching someone play a game, you should be respectful of their choices in how to play. Their style may be different from yours, but that doesn't mean that it's the worst approach in the world. Plus, you might learn new tricks you never thought could exist. It's also good to know when to give advice or suggestions. It's not good to do this when they appear to be doing just fine and are happy with their current progress. It's not good to say "You should do blank.. not blank,"but saying something like "Did you know that if you do blank, blank will occur?" Or if you've played the game before and want to give them a hint say something like, "When I played this game I found blank to be difficult, but I learned that if you do blank it's easier."
It can be fun to watch someone else play a game or have someone watch you play. It's all about being interested in the game and wanting to share your enthusiasm with your partner. Remember to always be nice to each other and have a good time.