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