Counting cards is remarkably easy, especially if you know a few tricks. Here's some that work for me.

1. Count the tricks, not the cards. If you know how many tricks of a suit have been played, and if anyone's thrown off-suit cards on those tricks, you know how many cards are out from a suit.
2. Remember there's only two kinds of cards you care about: those that are higher than your cards, and those that are lower. If you're trying to lose tricks in that suit, you only need to count lower cards; if you're trying to win tricks, you only need to count higher cards.
3. Some cards have equal value. For instance, if you have the 8, 9, and 10, all of those cards are identical. Treat them thusly.
4. Make your counting match your hand. For instance, if you have the Queen of Spades, you probably want to count every card in Spade tricks, but if you have just one Club, you don't need to count Clubs.
5. Make your counting match your strategy. If you're planning to dump the Queen of Spades on a Diamonds trick, you probably need to count Diamonds, but you can ignore Club tricks. If you're planning to lose the lead with a low Heart, you probably need to count Hearts, both to know for sure that you're playing a the lowest card that can, or is likely to, be played, and to be sure that there are other Hearts out there.

If you use the above tools, you won't have to keep track of all 52 cards, just the ones that are you need to watch to win!