The Dredge Recording Rarities pursuit is your classic 'sail to the four corners of the world to find the legends residing there' quest. In this case, those legends are four massive fish the travelling merchant asks you to catch, and for each one you show to her you'll get two Research Parts with which to unlock new gadgets for your boat. You can also sell each fish for around $250, meaning if you catch all four you'll be swimming in cash.
In order to complete Recording Rarities, you need to unlock explosives from Gale Cliffs, the Abyssal and Hadal rods from Stellar Basin, and you also need the Mangrove rod. I'd definitely recommend getting a better engine, too, since you'll be sailing to all four main areas on the map. Lastly, be aware that each of these fish take up lots of space in your cargo before you head out to wrangle them, but on the plus side, it doesn't matter if they rot or you sell them since you caught each and got the encylopedia entry to bring back to the merchant.
