Basic Mechanics
Many of the game's mechanics are explained in the tutorial, but there is plenty more that is left for you to find out for yourself. Some of the basic goings-on in the game are explained here in slightly more detail than in game. As of version 2.3, an official guide has been released here with information on these basic mechanics in detail.
Rank
Your rank granted at the end of a Cup is based on the point accumulated throughout the courses and prison stages. Points can reach a maximum of 550 for completing everything perfectly. Thanks to Niko Reiesu for explaining this:
• Reaching the podium at the end of a cup grants 50/75/150 points for 3rd/2nd/1st place overall.
• Total stage points earned through the GP grant some amount of rank points. A maximum of 60 gives 100 points.
• Breaking Prison Eggs will grant rank points up to a maximum of 100, individual eggs give some points based on the total number of prison eggs.
• Lap Bonuses give points, up to 100 for being in 1st every lap.
• Finishing a race with Rings in your tank will give 1 point for every 2 rings, up to 50 points
• Restarting or failing a stage will reduce your score by 25 points. Not using any retries grants 50 points.
The number of points obtained will give different ranks. A rank requires 425 points to obtain, and getting the Emerald on a Cup will upgrade it to an S rank. There are no challenges that specifically require S ranks.
Special Stages
To reach a Special Stage at the end of a Cup, an A rank is required. The Special Stage will start immediately after Round 5. Falling off the course during a Special Stage or letting the Emerald make it to the end of the stage without catching it will cost a life, and if all lives are used up, the stage will end, kicking you back to the podium. As of Version 2.2, some progress towards breaking the UFO Catcher is carried over after losing a life, but the console command "restartlevel" will no longer work to reset the stage without losing a life.

Notes
Heavy Magician
Some characters will face against Heavy Magician in Grand Prix races. When this happens, Heavy Magician will take on the stats (speed and weight) of your character.
Challenge Board
The Challenge Board is (mostly) randomly generated, though large and small size challenges will not change size. Don't rely on someone else's challenge board to find where a specific unlock is if you want to unlock it with a Key!
Chao Keys used to unlock Challenges can be obtained infinitely (or at least up until 999). One key is earned every 5 races as of Version 2.1.
Spray Cans are only in a set order for the first 9 unlocked, not counting those gotten with Chao Keys.