Around the map there are pieces of papers which give a general direction as to what you have to do.
There are two mechanics:
1. Spinning on the dance tiles
2. Moving your cursor/mouse on specific areas
There is also pressing the key "R" which when landed on red or black will take you to the corresponding world.
1st room in both worlds.
For the dance floor and soup, you need to spin on the dance floor W then A then S then D will count as a spin, and move your cursor on the soup for a specific amount.
2nd room in both worlds
For the 9x9 Grid puzzles you can press "I" on your keyboard, which will show the solution to the puzzle directly, the pieces of papers in those rooms hint at what the puzzles solution is.
Snail race
For the Snail race, it is a little difficult but you need to spin on the dance tile to advance your snail, it's a bit tough but after enough tries it should be possible to beat.
Rabbit race
For the rabbit race, you need to advance your rabbit by spinning on the rabbit head, spinning on the blue rabbit head will slow the blue rabbit.
Hopefully this helps!
If not just ask me where you are having trouble!