Game designers play games very differently than the average player: they play to learn. Designers study the games they play. They dissect the mechanics and try to understand the rationale behind the decisions the development team made when implementing them. Designers deconstruct games, peeling them apart piece by piece.
In this, game designers are not unlike cinematographers who learn about their art by watching and analyzing film. And, not unlike the cinematographer, there are some simple guidelines that can aid the game designer's study.
A Word of Warning
Before you begin down this path there is something you should know: playing games in order to study them is not what most people would consider "fun." This doesn't mean it isn't fun at all; it just means you have to think a different way.
You have to find joy in discovering mechanics and watching their emergent properties unfold. You have to be willing to endure a certain amount of tedium in order to glean clues about the inner workings of a game. Most of all, you have to be able to enjoy playing bad games as well as good.
Like the rest of game design "playing to learn" falls somewhere between a science and an art and contains all the joys of those two fields (though not many we traditionally associate with playing video games). If you can enjoy the eureka moments that happen when you finally discover how something is done, and the cascading flights of fancy that cause you to see the ramifications of a design that far exceed what's actually in the game, then this field is for you.
Start by Playing
Start by playing -- actually playing. I've seen designers lose the ability to play games and only retain the ability to study them. I'm going to go out on a limb and say that this is something you should avoid at all costs over the course of your career. I'll go even further and say you can't be a great designer if you can't play.
That said, designers rarely complete games. The designers I know all have to force themselves to play a game from start to finish. Why? After the wonderful explosion of possibilities is over and all the mechanics are in clear view, they say, "Oh! I get it ... " and move on.
There are too many games to finish them all and still go about making games; therefore you may have to sacrifice your completionist streak (and Gamerscore) in order to really play to learn.
After many years of playing, gamers develop a sense that tells them when something in a game is extraordinary. Those moments when you think, "Dang, that's cool," or say, "Hey! You've got to check this out!" are a response to this sense. Likewise, times when you say to yourself, "This is the worst pile ever. How could anyone have built this?" or just throw down the controller in disgust, those too are a response to this sense.
Usually, it's pretty vague. You might just get the feeling that something is either cool or unpleasant. Make a mental note whenever that happens because it's those moments that you'll want to dig into.
A quick practical tip to help you do this: Be aware of when you lose track of time -- that's where the great stuff is. Also be aware of when you put down the controller, when you decide to take a break and do something else. Those moments are not necessarily bad. In fact, they're often designed into games, but they are always interesting to examine.