Bronze

Academy Graduate
Earn 1 kill with each ship class (except the Support Frigate)

Red Shirt
Be the first to die in a Versus multiplayer match

Imperial Instincts
Earn 25 cumulative point captures in Conquest

Bridge Commander
Earn 250 cumulative kills in Team Deathmatch

Mirror, Mirror
Capture a point with the aid of your own wingman

Born Leader
Capture 3 points and help destroy the 4th in a single Assault match

Silver

NCC-1701
Earn 250,000 points as the Flagship class in one Team Deathmatch game

No Mercy
Earn 50 cumulative Escape Pod kills

Vulcan Wisdom
Don't die or shoot, yet capture 3 points (including the enemy base) in a Conquest match

Nero's Rage
Destroy 3 enemies while using a single invulnerability powerup

Gold

Starfleet Medal of Honor
No deaths or ejections, plus at 15 kills, in a single match

There is a secret trophy as well. The game has a demo up on the US PSN Store, but the Xbox version didn't get all that great reviews, so it seems you'd be better off skipping it.