Alternate Paint Schemes:
Get 3 S ranks with a vehicle to unlock it's alternate paint job. Alternately, just win 10 races with a car.
Unlock all cars' paint schemes.
Mirror Race mode:
Get an S rank on every track in Super Excite mode.
Crazy Super Truck:
Get an "S" rank on all races in mirror mode to unlock "The Crazy Super Truck". Crazy Super Truck has unlimited nitro.
Unlock Super Excite Mode then get passing ranks in every race to unlock the Diamond Cup with the Crystal Road track.
Thanks to Revolution Readers Manzai and The_Broose!