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