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