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