Successfully complete one of the following tasks to get a trophy:

Bag Blisterz Booty (Bronze): Bounty Blisterz Booty.
Starting the Journey (Bronze): Complete the tutorial.
Wash my hands of this mess (Bronze): Bounty Filthy Hands Floyd.
Clean the Water (Bronze): Bounty Looten Duke.
Spank that booty! (Bronze): Bounty Boilz Booty.
Dumb it down (Bronze): Rescue Eugene Ius.
Don't be talkin' 'bout ma mama (Bronze): Bounty Jo' Mamma.
Go Organic (Bronze): Bounty Packrat Palooka.
Junkyard Challenge (Bronze): Bounty Packrat Palooka.
Mine your business! (Bronze): Bounty X'plosives McGee.
The Sleg Avenger! (Bronze): Bounty Lefty Lugnutz.
You scratch my back (Bronze): Bounty Elboze Freely.
Too Much Fuzzle Pie (Bronze): Bounty Fatty McBoomBoom.
You know too much (Bronze): Take out D. Caste Raider.
Giving a Leg Up (Bronze): Get to Last Legs.
Dam Sekto! (Bronze): Enter the Lower Dam.
Elevate yourself (Bronze): Take out a Gloktigi.
Regain that spring in your step (Bronze): Reach the Upper Dam.
Now there's two? (Silver): Take out Sekto's personal bodyguards.
Smack down Sekto! (Gold): Defeat Sekto.
Back in the Village (Silver): Defeat the Wolvark attackers in the Native Village
Get 'em (Silver): Bounty 250 Outlaws
Get 'em goooooood! (Gold): Bounty 250 Outlaws alive.
Mo Moolah (Gold): Have 20,000 Moolah when you take the barge to Doc's Mongo Retreat.
Go with the Flo... (Bronze): Find a hidden bottle of medicine.
Crystal Mess (Silver): Collect 25 orange crystals.
Finders, Filchers (Silver): Collect 40 Moolah chests.
Free the Meat (Gold): Destroy the 10 hidden barrels.
Slakker Clakker (Bronze): Finish the game on Easy.
Stranger Danger (Silver): Finish the game on Normal.
Sekto Suicide (Gold): Finish the game on Hard.
Farmer Harmer (Bronze): Kill Farmer Beeks.

Additionally, there are five secret trophies:

Secret trophy (Platinum): Unknown.
Secret trophy (Silver): Unknown.
Secret trophy (Silver): Unknown.
Secret trophy (Bronze): Unknown.
Secret trophy (Bronze): Unknown.