Here is the full list of all 11 Gem Worlds achievements.
Complete the fifth world.
Complete the fourth world.
Complete the third world.
Complete the second world.
Boost your light by 50% or more.
Complete the first world.
Hammer 10 monsters in one level.
Kill all monsters in a level.
Keep 10+ items in your inventory at once.
Complete a level without taking damage.
Collect every gem in a level.