Generally, you'd probably want to have 2-3 matching moves for a contest, and at least one that's against type. So for example, a Clever move in a Cool contest. Why? Well say there's only 2 stars left to fill up the judge's bar, but it's your turn. If you use a Cool move, and the next guy does, you just gave him a full boost. You want to avoid that by making the judges upset and lowering the number of stars. The best move is one that tampers with everyone's scores, too.
Now, if you do reach 5 stars on your turn, your pokemon mega-evolves, if able.
Some moves kind of act like combos together even if they don't technically double your score. For example, a move that makes you move first, and a move with a higher score when you go first. Accupressure doubled scores no matter what you used next in Gen 4, but I don't have a pokemon with that move yet so I don't know if they kept that.
Finally, some of these combos are different from Gen 3. Sing, for example, used to combo with Perish Song and Refresh.
Sorry for the shitty handwriting.
