Accomplish the indicated achievement to get the corresponding number of Gamerscore points:

Adventure is Out There (15 points): Defeat the first wave of combat planes.
To Paradise Falls (15 points): Complete the level to start Carl's journey to Paradise Falls.
A Bird Named Kevin (15 points): Find Kevin before Muntz does.
Slide and Slither (20 points): Defeat the giant Anaconda.
A Voice In The Mist (20 points): Find the voice in the mist.
Meet the Pack (20 points): Meet the pack: Alpha, Beta, and Gamma.
Night in the Jungle (20 points): Avoid being captured by Muntz's dogs.
Sinkhole Terror (30 points): Defeat the giant Crocodile.
Paradise Falls (30 points): Help Carl get his house to the top of Paradise Falls.
Spirit of Adventure (30 points): Free Russell and Kevin, then escape the Dirigible.
Adventure is Here (50 points): Defeat Muntz's canine squadron in the skies above Paradise Falls.
Teamwork Badge (30 points): Defeat either the giant Anaconda or Crocodile with 2 players.
Wings Badge (10 points): Defeat the first plane in the battle over Paradise Falls.
Animation Badge (20 points): Unlock the Bonus Art.
First Aid Badge (10 points): Use Russell's Canteen three times.
Bug Hunting Badge (50 points): Catch or squash 100 Dio Bugs or Butterflies.
Bird Watching Badge (10 points): Locate Kevin for the first time.
Defense Badge (15 points): Successfully defend against 10 dog ambushes.
Fishing Badge (20 points): Survive against the hungry Piranhas.
Global Citizen Badge (20 points): Make sure to put out your campfire.
Hiking Badge (10 points): Find the picture-perfect view of Paradise Falls.
Entomology Badge (20 points): Collect all varieties of each Butterfly and Dio Bug in the game.
Karate Badge (20 points): Break open 10 watermelons.
Kite Badge (20 points): Use Carl's House to float across all the chasms in the game.
Sharp Tools Badge (15 points): Use Russell's key to cut through three items in the game.
Knot Tying Badge (10 points): Tie off the house so it doesn't float away.
Music Badge (10 points): Use Russell's Bugle three times.
Recycling Badge (15 points): Play through any previously completed level in Story Mode.
Shoveling Badge (25 points): Dig up all of the hidden bones using Dug.
Swimming Badge (10 points): Swim through all of the areas that have water.
Survivor Badge (50 points): Complete any Story Mode level without failing or quitting.
Artifact Curator (35 points): Find and collect all of the Artifacts.
Memorabilia Specialist (30 points): Collect all of Carl's Mementos.
Merit Badge Champion (50 points): Collect 6000 Merit Badges in the game to become the Merit Badge Champion.
Buddy System Badge (25 points): Free your buddy from five Snare Traps.
Environment Badge (20 points): Break apart 500 objects.
Pest Control Badge (25 points): Squish 10 Wasps and Spiders.
Multiplayer Extras (50 points): Unlock the extra Multiplayer Combat Maps.
Ellie Badge (50 points): Complete all of the Quest Cards for a special badge.

Additionally, there are three secret achievements:

Rare Statue Badge (25 points): You've found the Gold Jaguar!
Excavation Badge (25 points): You've found the hidden cave in "A Voice in the Mist."
Beginner's Wings (10 points): You've accessed the Multiplayer game menu.