Choosing the three cards to pass at the beginning of the round is a key part of Hearts strategy. This is the best opportunity you will have to improve your hand, especially by getting rid of cards that could be problematic for you.
After the pass, the only way to get cards out of your hand is to play them, so if a card is too dangerous to play, it's best to pass it.
In spite of the game having the name of "Hearts", the most important suit in a game of Hearts is actually spades. Why? Because the Queen of Spades is worth 13 points, which is as much as all the hearts combined. Avoiding the Queen of Spades (when not shooting the moon) is your primary goal each round.
Because of the Queen of Spades, the three most dangerous cards in the deck are the Ace of Spades, King of Spades, and the Queen of Spades itself. These are the three cards most likely to result in the person who plays them winning the queen, and thus suffering the 13-point penalty. Generally speaking, these three cards are problems that you should pass on to someone else.
First, though, count the spades in your hand. If you have four or more, spades is a reasonably "long" suit for you, which means you have enough of them not to worry about your high spade. In fact, if you have at least four spades, it's usually better to hold onto the queen rather than passing it. This way, you can control what trick the queen falls on, and hopefully, dump it off on someone else.
Any spades lower than the queen should always be held onto.
While not as dangerous as high spades, hearts is the other suit that is full of points. Low hearts are actually quite valuable since they will keep you from winning tricks full of four hearts. High hearts (jack or better) should generally be passed for the same reason unless you are long in hearts.
Clubs and Diamonds
High cards here aren't as dangerous as the other suits, but they're a lot less dangerous if you have low cards to back them up. If you only have one or two cards in one of these suits, you should pass them both, in order to create a void suit. By getting rid of all your cards in one suit, you can get rid of your trash whenever that suit is led.
Aces are the best cards to pass to the left. The reason for this is that aces are the most likely cards to win tricks, and you want the player to your left to win tricks. If she does, she will lead the next trick, allowing you to play last in the trick, so you can see exactly what to play.
The Queen of Spades is a good card to pass to the right, because the player on your right always plays before you unless you lead the trick. This means that on any trick you don't lead, you will know before you play your card whether the Queen of Spades can possibly fall on the current trick. This is a tremendous advantage.