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