Here is the full list of all 52 Gem Rush achievements.
Score 20 points within the first 7 turns of a game
Score 20 points within the first 6 turns of a game
Score 15 points within the first 5 turns of a game
Enter 5 distinct rooms in a single turn
Perform 10 different Feats
Build 3 rooms in a single turn
Build a 5-point one-color door
Perform a construction where you score at least 2 points per card used
Use all your cards on construction, then gain at least 3 new ones before your next turn
Build a 6-point door
Score 2+ points on each of 3 consecutive turns
Score 3+ points from constructions in a single turn playing only Diamond Dust
Perform a construction where you score at least double the door's normal value (and more than zero)
Perform a construction where you utilized every gem you played
Score 3 points on your first turn
Score 4 points on your first turn
In a single turn, build a room, and then build one of the doors in the newly-constructed room
Score your first point
Score 100 points across any number of games
Score 400 points across any number of games
Score 1000 points across any number of games
Score 2500 points across any number of games
Score 5000 points across any number of games
Finish a Crisis Mode game with a combined score of 35 or more
Finish a Crisis Mode game with a combined score of 45 or more
Finish a Crisis Mode game with a combined score of 55 or more
Finish a Crisis Mode game with a combined score of 65 or more
Finish a Crisis Mode game with a combined score of 75 or more
Play a game in each mode
Play a game with each number of players
Score 5 points on your first turn
Score 6 points on your first turn
Score 7 points in a single turn
Score 8 points in a single turn
Score 9 points in a single turn
Score 10 points in a single turn
Score 20 points within the first 5 turns of a game
Perform 20 different Feats
Finish a game where the combined cost of the doors you built was lower in your player color than any other color
End a turn with 7 or more cards in your hand
Finish a game where the combined cost of the doors you built was higher in your player color than any other color
Score 4+ points on each of 2 consecutive turns
Build a 6-color door
Build a door costing 2+ gems while keeping 3+ cards in your hand
Use 4 cards in a single construction
Reveal and keep 4+ cards when using a room ability
Score 20+ points in 3 consecutive constructions
Play an online game
Play 3 online games
Play 9 online games
Play 27 online games
Play 81 online games