Here is the full list of all 33 Star Trek: Bridge Crew trophies - there are 15 bronze, 13 silver, 4 gold and 1 platinum.
Destroy 20 Klingon ships
Complete a mission as Tactical
Disrupt systems 5 times via a system intrusion
As Tactical, scan your first ship
Witness the death of an NPC crew member
Complete a mission as Engineer
Destroy 5 non-Klingon ships
Destroy a ship using only torpedoes
Survive a battle against three or more hostile ships
Successfully transport 30 people
Complete a mission as Human
Play 6 missions with friends online
Complete a mission as Captain
Perform 5 warps as a Helmsman or Engineer
Navigate through an asteroid field unscathed
Complete an Ongoing Voyages mission
Complete a mission as a Vulcan
Complete a mission as Helm
As the Helmsman or Engineer, perform your first warp in a Campaign or Ongoing Voyage mission
Complete a task in the training scenario
Obtain all the Trophies.
Finish a mission without using torpedoes
Finish a mission without using the shields once
Finish a mission using the original bridge
Complete all campaign missions and one of each Ongoing Voyages type
Visit every star system in The Trench
As the Engineer, Repair 15 systems
As the Engineer, maximize engines by routing power from shields and phasers
Complete a mission with less than 10% Hull Integrity
While on a mission, look outside of the spaceship using Space View for 10 seconds
Answer your first Hail
Rescue at least 120 people in the stand-alone Kobayashi Maru mission
As Captain, temporarily take over an NPC crewed station during a mission