Minigames start at XX:05 and XX:35 (01:05, 01:35, 2:05, 2:35, 3:05, etc.)

After surviving all rounds or reaching the finish line in, you will get Kay Tokens that can be used to spin the wheel in Queenstown. You will also get a buff that lasts for 15 minutes if you win, with the buff healing you for 1% of your maximum health every 2 seconds.

OX Quiz

You will spawn in the middle of the map, and be given a few seconds every round to stand on either platform to the side to answer the question that pops up. O is for True, and X is for False. Don't stand in the center when the timer runs out, because barriers will appear, preventing you from changing answers, and you will fail the minigame.

Miyu#9404 on Discord has made a very useful spreadsheet with a collectin of questions and answers here.

Speed Runners

A very easy minigame. It's probably the best one for you if you want the guaranteed Kay Coin for completing/winning a minigame. There are many shortcuts and traps that you should watch out for.

Here is an example of me playing Speed Runners (definitely not the most optimal run) if you just want to see what it looks like:

Spring Beach

This is a minigame where certain blocks on the floor will shake before shooting anyone on top of them to the sky. To make this minigame easier, you can go to your settings and hide other players and their name tags. Check out Settings for how.

Ludibrium Escape

Your goal is to reach the very top floor within 3 minutes. The game requires precise timing on your jumps, since the blocks will be moving, and there will also be destroyable blocks that you will have to clear out of your way. A small tip is that when jumping across a gap between two blocks that are on the same altitude, and there is a block right above you, you can jump a little early, and even if you bump your head against the block above you, you will still move upwards once you have moved onto the gap.
Saintone's's Ludibrium Escape Guide:

Dance Dance Stop

In this minigame, there are platforms where a number will appear. If there are exactly that many players on that platform, it will not disappear when the countdown ends. When there are enough players on the platform, you can interact with the panel to create a barrier to prevent people from entering or leaving. There will also be bonus rounds where you go to a corner and interact with the circle things to start dancing, giving you bonus experience. If you need more people on your platform, try saying so, so that the people who are wandering around can join you.