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