This quest is given by Hoggle Bogstrutter after you complete the Scattered Notes quest.
The first thing you will need to do is read the entire Bogstrutter Isle Guide.
Visit a number of landmarks on Zek. This is similar to the Tour of Antonica/Commonlands quests.
After each landmark, you examine the book for information on the next.
Bogstrutter Cove: 503,262
Greenspire Wood Inn: 337,-20,-465 or 637,-17,22
Rustic Pond Resort: Near 293,-20,-350
Rodcet Steam Hall: 687,-17,-323 inside of the caves
Celestial Abyss: This is just down the elevator at the Orc Mines 343,-114,-118.5
Windspirit Round: Druid Ring 228,14,93
Spiralball Arena: This is the spirit lake with the spiral path and the NPC in the middle -58.5,-3.65,-71.6 . VERY IMPORTANT not to step in the water or you will spawn a large number of monsters who will surely kill you.
Pirate Bay: 146.82,-44.57,396.31 on Siren Island
Tower of the Titan: -23.71,-47.11,-351.56 on top of the large rock
Castle: -162.16,-25.11,-615.68 out front of the Deathfist Citadel
After visiting them all return to Hoggle Bogstrutter at the docks for your choice of breastplates. This also opens up the Tour of Feerrott quest.