I'll use odds at different times depending on the type of game I'm playing. I feel it's important to note that calculating odds is just one part of the game.
In cash games, I'll consider them almost every hand. In a tournament, I feel there's a point in the game -- assuming you've held in there long enough to build a decent stack -- when it's time to start playing the odds. Before then, my goal is to accumulate chips. So, I play relatively loose and aggressively in the beginning trying to control the betting and stealing pots; at that point in the game odds aren't really relevant to me.
That's just my personal style. Some players are very meticulous about playing the odds all the time, and it works for them. In any case, understanding odds is something you should definitely shoot for. The more you know about the game, the better you'll play.