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