Bronze

Big Red Button
Go ballistic… Use your first nuke!

Supaweapon!
Collect components to complete a megalaser.

Joint Command
Stand together with another player to complete one mission.

Comsat Uplink
Play online with three other players.

World Tour
Take a tour of each red star country and trash the place! Complete the first 9 missions.

Cockroach Of Freedom
Withstand eight minutes of the red star onslaught in any survival game.

Cannon Happy
Complete 9 missions without nukes or reinforcements.

Champion Of Freedom
Survive 12 minutes of the enemy's attack in any survival game.

Massive Destruction
Destroy 10 enemies with a single nuke.

“Ow, Quit It!”
Destroy the supply chopper in mission mode. Who needs power-ups?

Silver

Stockpile
Amass 5,000,000 points.

Ultimate Tanker
Dispel the enemies of liberty. Finish mission mode.

Gold

Freedom's Last Stand
Deafeat all 19 red star champions in boss blitz mode.

Source: I got the trophy list from Kryomancer's trophy list