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