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