The real answer is: enough to give your player a smart opponent, and as many as you can get away with whilst giving your player a high quality experience.
But that's probably not going to help yet since you can't really calculate moves in any decent amount of time.
What it's going to take to make a smart AI
You can't afford to evaluate every possible result of every possible move. Only seeing two moves ahead makes for a slightly unintelligent AI in a game where it's often said the player who can see the farthest ahead will win. Former world chess champion Garry Kasparov is quoted here:
Asked how many moves ahead he can think, Kasparov replied that it depended on the positions of the pieces. "Normally, I would calculate three to five moves," he said. "You don't need more.... But I can go much deeper if it is required." For example, in a position involving forced moves, it's possible to look ahead as many as 12 or 14 moves, he noted.
So 3 ahead is the minimum, and it's clear you can barely calculate farther than that before you reach insane wait times between turns, thanks to the exponential nature of the calculations.
You need to optimise.
Alpha-beta pruning works to reduce the amount of possible paths you evaluate in a min-max tree. It cuts out paths that lose in the long term and identifies paths where wins are possible so you don't waste time on fruitless searches.
Even with AB pruning your AI will never be able to evaluate all possible games in a timely manner for any significant number of moves. You need to set it a time limit - say 5-10 seconds, maybe more in situations where the computer is really having trouble finding a good move. Once the time limit is up (if not sooner) your computer must stop and make the most appropriate choice based on what it worked out.
Note that the most appropriate choice isn't necessarily the best one (for its winning), since sometimes you might want your AI to stuff up a bit. Note also that in its limited available time, your computer may not actually find an optimum move, or even necessarily a very good one.