Lemme break this shit down for you folks.
The main concept I discovered was that you could loop around the finish line because there are no checkpoints in this game. The game only knows when you are in bounds so if you go out of bounds and loop around back behind the finish line it thinks you went all the way around the course so the lap counts up. The key is being able to move in and out of bounds so stick too far out of a fence or wall then you won't be able to go back outside the course to do a shortcut again.
Main Concepts
Amy - She boosts through walls or just wiggles through them. Use her boost to go faster.
SuperSonic - He just falls through walls with pure speed or after bumping them a few times.
Sliding - Hold either left trigger or right trigger to turn sharper.
Stopping - Hold both triggers down to stop the character asap, like Amy boosting after breaking through a wall or SuperSonic to stop quicker to turn around.
3lap - Do as quickly as possible with looping after getting out of bounds. Requires quick turning which involves tapping b (gas on the game cube so to speak) very quickly so you just walk in a tiny circle. If you go back in bounds you are screwed.
flap - Get Amy or SuperSonic nearly perpendicular to the finish line without crossing it. Move up very close to it and then start looping. Unlike a 3lap, you can perfect your set up before starting the next lap. With Amy, you use her boost to finish quicker. SuperSonic on RC involves tapping the triggers quickly so he slides out in an even smaller circle or you can skid across the finish line, go out of bounds and then circle back around.
RI - looping
Amy -
Either boost through the rock, slower/easier, or wiggle through it by boosting towards it, faster/harder. Then you just loop around the fence but the part where the game sees you out of bounds is a little beyond the fence.
SuperSonic -
Inferior for the 3lap but I make the breakthrough look awesome. He is good for the flap too.
RC - looping
Amy -
You can fall out of any side of the poles but I think this one is better. You have to let the game know you are back in bounds so that is why I head towards the road after getting down there. Rev is tricky because you just have to do it from the other side but not that complicated to figure out.
SuperSonic -
We seen this one a lot, supersonic does it the same way and is best for the flap.
RR - looping
Amy - Only good when the snow is turned on in GP mode. There are those chamber things to your right at the start that you can drive into. Go inside the one farthest to the right and you can go out of bounds in the right corner of it using Amy's boost. Makes the flap a little easier to do, inferior for 3lap. You can do it without the snow, just really hard.
SuperSonic -
Just got to get the feel for how to hop inside there. It becomes quite easy with practice. You are looping in a very tiny area so be careful! If you can keep supersonic from falling into the water that would be the best. Luckily, Sonic can't drown in this game.
RF - looping
Amy -
You can just wiggle through which is better or opt to boost through the wall which is easier. If you are lucky, you fall through the wall to your left where I am boosting which gets you inside much faster, just need lots of luck.
SuperSonic - You just wiggle around a lot in that spot where you go through the wall with Amy. If you are very lucky you might just pass through it or just after a few bumps into it. Potentially faster but very hard to consistently do, also okay for flap.
RE - only course without looping
Amy -
There are three variations here, the one I do, instead of falling out of the level in front of the right pole like I did do it on the other side, and falling out of the level through the left pole so you are behind the finish line so facing forward in gp/ta mode. For the 3lap, you got to hit it 3 times so that sucks. You just got to play around with it a lot until you get lucky. The left pole is faster for gp and ta mode but much harder to fall through. The rev flap is extremely hard because you have to fall out of bounds to loop around to the part ahead of the loop, similar to how supersonic gets back on the track in his spot in rev 3lap. Be very patient.
SuperSonic -
At the end, I am sliding on that wall. If you are lucky, you can fall through it and bypass the loop to finish the lap faster. I have fallen out in some other weird places but it is very rare so the more you play the better your chances. Could be done 3/3 but its random.
Since it is very random where you can fall out, some of these records can be done still faster so if you are bored give this game a try and see what more you can discover!
Have Fun!!!