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