Here is the full list of all 21 Volcanon achievements.
Enter the World 1 stage select.
Enter the World 2 stage select.
Enter the World 3 stage select.
Enter the World 4 stage select.
Clear the game.
First time to get a crest.
Do 100 magma jumps.
Do 1000 magma jumps.
Do 10000 magma jumps.
Get three cores and clear the stage with no damage.
Defeat the World 2 boss with no damage.
Defeat the World 3 boss with no damage.
Defeat the World 4 boss with no damage.
Get all crests and cores.
View clear results with all cores in possession and a total clear time of 30 minutes or less.