Extra Car:

When playing in arcade mode choose any car. Then
play all tracks in order. You must make a record
and at the last track you will see the staff names. Then you'll get a new car named Daytona.

Dinosaur Canyon area:

Turn the car around at the starting line, then drive up the ramp on the right.

Daytona bonus car:

Finish in first place on all tracks in a normal
race under the normal difficulty setting.

Uma horse:

Finish in first place on all tracks in a normal
race under the hard difficulty setting.

Uma 2 horse:

Finish in first place using Uma on all tracks in
a normal race under the normal difficulty setting