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