Divine intervention is way better when used when you got one card of each color to kill a monster. That give you +2 in card advantage, and the aim of the game is to lose the less number of cards possible.

Don't underestimate the power of Support someone else. Not only you gain half a turn if that player is in another place, but you make two players around the table know which card someone have (which let you plan around it). But usually only give cards until he got 5, else he will need to discard the extra. When you have only 2 or 3 cards remaining, drawing more will make only draw 1 or 2 extra cards, which isn't great. Instead, if you can give them to someone else, that let you draw 4 next turn.

Close to the end, try to kill all monsters from 4th to 6th close to the same timing, if only one monster is remaining, 60% of the cards of the deck will not be much useful.

When playing cards, start to play cards of a color not already done in a monster. Then color not present in other monsters.

If you are alone in a monster, close to kill it, and don't have the right color to kill it, try first to give your hand to someone else. If you don't want to, you could try to consult the oracle, since the timing of it should be weird, that tell others you don't have the color. So if you need two more cards to kill it, and have two cards on hand, it's a direct call to give you 2 to 3 cards to make you kill it.