Like all trick-taking games, the gameplay is all about who has control. The difference is that you need to control whether you're leading or following, that you may want to win or lose.
Watch for when you or your opponent crosses 4 tricks. That determines who will aim for high and who will aim for low. It's to your advantage to not reach 4 tricks first.
If you reach a state where each of you has 4 tricks, try to take as many tricks as you can.
If you think your opponent is going to go low, spend the first part of the round by letting them win tricks (play low cards). After they've crossed 4 tricks, use the Fox and Woodcutter to increase the strength of your remaining cards.
Your goal should be to get 6 points each round, whether that's through winning few tricks or most (not all) tricks.
Examine your cards and determine how many tricks you could win. For example, if you have 9, 10, and 11 of the trump suit, you are guaranteed to win 3 tricks. The remaining cards determine whether you should go low or high. The Fox complicates this, as the trump suit can change.
If you are going low, pursue that goal early. If you are going high, try and set up a way so that your opponent has to take the last few tricks (so you don't take them all).
Try to play the suit you have the least of. Before you play the last card in that suit, try to make that suit trump. For example, if you have 3 Moon cards, 5 Temple cards, and 5 Key cards, play all but one of your Moons, then change the trump suit to Moons with a Fox. Your opponent likely has more Moons than you at this point. So now that you have no Moons, you can play any suit, which makes control easier.
If half your cards are of rank 6 or lower, try to lose all the tricks you can. Get rid of 9s as they will turn into trump cards.
If half your cards are of rank 6 or higher, try to win all the tricks. Foxes let you switch the trump when you run out. Use 1s and 2s to let your opponent have some tricks so you don't get stuck with 10 or more.
Strategies By Card
Swan (1) - Lets you maintain control if you want to lose tricks. Good for maintaining control of the game (especially if both opponents are going for the same thing (low or high)).
Fox (3) - Use to change the trump suit to one you have a majority of cards in (if you want to win tricks) or a card you have few or one in (to lose tricks). Good for maintaining control of the game.
Woodcutter (5) - Play anytime to get rid of a card in your hand that doesn't fit your goal. For example, if you have a low card and you want to win more tricks, or a high card and don't want to win more tricks. Good for if you need to change strategy in the middle of a round.
Treasure (7) - Useful only as a response to lower cards, otherwise the opponent will win the trick and gain an additional point.
Witch (9) - Useful as a response to higher ranked cards of the same suit (unless the suit is trump). Hold onto the 9s until the 10s and 11s are played.
Monarch (11) - Useful to get your opponent to discard a card from the same suit. Useful if you have some mid-range cards in that suit (5, 6, 7) to win with.
