Always look ahead for your next move. If you run out of valid moves, the game is over. Check carefully to see whether your move will open additional match options, reduce your available matches, or leave you without any further matches at all.
Matching a set of gems on the bottom of the screen will definitely affect all the gems above it, so if you can see two valid matches, consider the top one first.
Combos and cascades score extra points! The more combos or cascade matches you can make with a single move, the more the bonus points rack up. Look for moves which will either make multiple matches at once, or cause the gems above to fall down into more ready-made matches.
Matching four gems will create a flame gem. Using the flame gem in a match will cause it to explode, detonating all nearby gems. The catch, of course, is that the more gems you remove at once, the more difficult it is to predict what the board will look like after the explosion. Be careful not to remove all your match options in one fell swoop! Unused flame gems carry over between levels, so if you're not sure whether using one would be a good idea, just hang onto it for later.
Making a T or an L match of gems will create a star gem. Using a star gem in a match will destroy all the gems in the row and column that the star gem is in.
Matching five gems creates a hypercube. If you match a hypercube, it will remove all the gems of the color you swapped it with!