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