Here is the full list of all 13 Archaeogem achievements.
Complete the Prologue chapter
Complete the Crystal chapter
Complete the Jungle chapter
Complete the Underground chapter
Complete the Epilogue chapter
Collect all the emeralds
Show 'em your dance moves!
Give your legs a little rest
Dash through 5 green flowers without touching the floor
Find Rose's lost flower bouquet
Give lettuce to the Turtle
Reunite the mushroom boy with his father
Collect all prismatic stars