Here is the full list of all 35 Fallout Shelter achievements. It takes around 80-100 hours to unlock all of the achievements on Windows.
Collect 20 Legendary Dwellers
Complete 100 objectives
Craft 1 of every room theme
Change the appearance of 10 NPCs in the Barbershop
Collect 20 Legendary outfits
Complete 100 Quests
Collect 20 Legendary weapons
Build 1 of every room type
Upgrade 20 Rooms to Level 3
Craft an Outfit, a Weapon and a Theme
Complete 60 Quests
Collect all Pieces of 4 Themes
Scrap 500 items
Craft 10 Weapons
Craft 10 Outfits
Successfully complete 50 Room Rushes
Collect all Pieces of 1 Theme
Level up a Dweller to level 50
Collect 100 Stimpacks and 100 Radaway on Quests
Complete 30 Quests
Make 25 Dialogue Choices in Quests
Kill 500 enemies on Quests
Complete 10 Quests
Kill 10 Quest Bosses
Level up a Dweller to level 25
Make 25 babies
Collect 10,000 Caps
Level up a Dweller from 1 to Level 10
Build 50 rooms
Build 25 rooms
Merged 3 Food Rooms
Merged 3 Water Rooms
Collect 1000 Caps
Merged 3 Power Rooms
Successfully stop 50 Raider Attacks