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