RTA Guide/The Secret Aquarium
Make sure you always give full input with your joystick during the turns so the curve is as tight as possible.
For the first three of the lower reds you can line up Mario like this to get a good turn.  For the very last red coin you can line up Mario like that to get a good turn. This allows you to turn really early, as in the video below. 
You can reduce lag at the beginning by buffering R, C-Up, and A, and you can reduce lag at the end by using a claw-grip to hold C-Left and C-Up until you grab the star.