Here is the full list of all 25 Mercury Race achievements.
Get the Gold Badge in Road 1-1
Get the Gold Badge in Road 1-2
Get the Gold Badge in Road 1-3
Get the Gold Badge in Road 1-4
Get the Gold Badge in Road 1-5
Get the Gold Badge in Road 2-1
Get the Gold Badge in Road 2-2
Get the Gold Badge in Road 2-3
Get the Gold Badge in Road 2-4
Get the Gold Badge in Road 2-5
Get the Gold Badge in Road 3-1
Get the Gold Badge in Road 3-2
Get the Gold Badge in Road 3-3
Get the Gold Badge in Road 3-4
Get the Gold Badge in Road 3-5
Get the Gold Trophy in Galaxy 1
Get the Gold Trophy in Galaxy 2
Get the Gold Trophy in Galaxy 3
Get 5 Perfects
Get 10 Perfects
Get 15 Perfects
Destroy the ship during a race
Get 5 Ultracombos during a race
Get any Badge without taking any hit
Get any Badge with 100% enemies destroyed