To complete the main path of Story mode, you only need to play a handful of levels on each planet. Below I have listed all of the levels you need to beat in order to earn this trophy, and the trophies you will earn along the way.
Planet 1 - LittleBigPlanet 1) Karting Lessons - tutorial
2) Garden Grip - versus race
3) After The Wedding - versus race
4) Serpent's Shrine - versus race
5) Mine the Gap - versus race
6) King's Castle - battle arena
Completing all of the above-mentioned levels with earn you:
Planet 2 - Monster Islands1) Turtle Island - versus race
2) The Emporer Has No Clues - versus race
3) Huge Monster Rally - versus race
4) Night Rider - versus race
5) Egg Kartin - grab 'n' run
Completing all of the above-mentioned levels with earn you:
Planet 3 - Victoria's Laboratory1) Sugar Rush - versus race
2) Current Events - battle arena
3) Cakes on a Train - versus race
Completing all of the above-mentioned levels with earn you:
Planet 4 - The Progress Emporium1) Future Perfect - versus race
2) Zeppelins Rule! - battle arena
3) The Infallible Breakfast Machine - checkpoint arena (vs. AI)
4) World's Fair in Love and War - versus race
5) Best Before Date - checkpoint race
Completing all of the above-mentioned levels with earn you:
Planet 5 - Eve's Asylum1) Roots Of All Evil - battle arena
2) Firebug Circuit - versus race
3) On the Wormpath - boss battle
Completing all of the above-mentioned levels with earn you:
Planet 6 - The Space Bass1) 2.0 Bee Or Not 2.0 Bee - versus race
2) RoboBun Test Chamber - battle arena
3) Huge Spaceship - versus race
4) Lost in Bass - versus race
5) Drum Smash - versus race
6) The Funkhole (and Beyond?) - checkpoint race
Completing all of the above-mentioned levels with earn you:
Planet 7 - Hoard, Sweet Hoard1) The Garage at the End of the Craftverse - boss battle
Completing the above-mentioned level with earn you both of the following trophies: