what article should go before specific games?

This is a good question, because it's difficult to answer. Articles are either for definite or indefinite things.

If I want to refer to a specific game, I would use a definite article: I am going to the game.

If I want to refer to a game, but it isn't clear that it's a specific game, I would use an indefinite article: I am going to a game.

In the first example, it is either clear which game I am talking about or I have already indicated which game in some other way.

Likewise, if I'm at home, and I make a mess on the floor, I might ask someone to get me the broom, because we only have one broom and we all know which one I'm talking about. Or, I could point to it and say, can you get me that broom? If, however, I am at someone else's house, and I make a mess, I may not know whether or not there is a broom, so I would ask, do you have a broom, or, can you get me a broom?

In the case of sports, these are not generally modified by articles:

I am playing cricket is correct. I am going bowling. I am watching baseball. I like football. Soccer is my favorite sport. However, I would say the soccer ball or the football or the baseball if I am referring to the ball itself. If there is a sport with a name that includes an article, then I suppose you would in that situation also use it, but I can't think of one.

A sentence like "I am playing the cricket" is not wrong, actually, but I would regard it as a bit of an affectation. I think in British English it might be more common than in American English, so it would be alright to say "I'm going to watch the cricket" though in America we would more likely say "I'm going to watch the cricket game." Likewise, we in America wouldn't say "I'm going to play the cricket game" though we might say "I'm going to play in the cricket game" if that's the pertinent piece of information. Instead, we would say "I'm going to play cricket." Since you're asking about cricket, though, perhaps you are indeed asking about British English. In that case, I'd have to defer to a Brit, though I do think the verb "play" may be less likely to be used in this case than "watch."

Sometimes, in America, we might do this to be funny (since British things are funny to us and not in a mean way—we genuinely find British people hilarious), but it's not normal usage.

If you want a definitive answer, "I am playing cricket" is always correct no matter what, whereas "I am playing the cricket" or whatever sport may or may not be acceptable depending on the context, whether you're talking or writing, etc. "I am playing a cricket" would never be correct, since cricket is always a specific, definite thing, even if it's abstract (there aren't more than one sports of cricket like there are more than one apples you could eat, so you could eat an apple if it doesn't matter which apple you're eating or you could eat the apple if you're referring to a specific apple or happen to be in the Garden of Eden).

Note also that we sometimes stress the word "the" if we're really trying to hammer it home that we're being specific. In that case, we would even pronounce it "thee" instead of "thuh" but it depends:

"Are you watching the Super Bowl?" "No, it's just a game." "Are you kidding? It's not just a game. It's the game!"

To make things more complicated, you could even stress the "a" above with a longer "uhh" sound if you're really serious. I hope this answer didn't muddy the waters or the cricket pitch too much for you!

I am playing Cricket

is the correct sentence.

If you want to use "the" in the sentence, you can use it like,

I am playing the game of Cricket

If you want to use "a" in the sentence or if you mention the format of the game,

I am playing a cricket match. (or) I am playing a test cricket.