So I can mouse navigate the menus, but I'm not sure how to make the cards move? is that normal?
Yes, I'm sorry the game is lacking in giving feedback to the player.
To play you need to click on the card in your hand to select it (it should move up), then click on a card on the field that matches with it (no feedback here, sorry, but try to find a matching flower or plant, or use a hanafuda suit cheatsheet).
If you cannot or simply want to add your card to the field then click on an empty space (But it will not work if the card has a matching pair on the field).
You then have to play the card on the top of the deck again by matching it if possible or add it to the field.
When you score a combo, a popup appears showing you the score you can make and you can either end the round to collect the points, or call KoiKoi to keep player and try to score higher but risk loosing to the opponent.