A recent update has made cherished trophies much quicker/cheaper to purchase.
To purchase a cherished trophy go to the Shipwright Shop at any Outpost. You have to be Class 30 (previously 100) in a Pirate Milestone to unlock the Cherished trophy. You unlock the Admired, Prized, and Treasured trophies at class 5, 10, & 20. You must buy all 3 of these before you can buy the Cherished trophy. It’ll cost a combined 300k gold to buy everything. (It used to be a million.)
You need to be aboard a Captained (owned by you, or a member of your crew) ship in order to progress towards Pirate Milestones. Some of them you will reach class 30 through natural progression. Playing for 150 in-game days, or setting foot on a different island 300 times will get you to class 30 in that Pirate Milestone.