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