Buy one fish of each species
There are a total of 18 species of fish you need to buy for this achievement, which includes the sea horse and the sea turtle. You'll unlock species as you progress, with the sea turtle being the last one to unlock, at level 48. This is the longest grind in the game, getting to level 48. The "Good student" achievement solution has more information on leveling up. Once you reach level 48 you'll unlock the sea turtle. Purchase one of each to unlock this achievement. Keep in mind that you don't need to own all of the species at once. Once you've bought a fish, you can sell it and use the money to buy other fish.