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