The Beachcombing questline is a four-part quest given to us by Magnus as soon as we complete Bayoo's Rescue the Pirate Girl quest and save Tanya. At the same time, you'll also be given a quest by Tanya herself, called Preppin' for Booty.
Part 1 of 4
Get 3 Pickaxes
Mine 3 boulders in the Cove
Make 2 brand-new Seaside Cups
Note: Chances are good that this first part of Beachcombing and the second part of Tanya's Preppin' for Booty quest will coincide with one another. If they don't do this naturally for you, however, you may want to force it, because there are so many overlapping elements that will satisfy both quests.
Part 2 of 4
Make 2 Octopus Bowls
Expand to Myra's Ocean Outlet
Sell 6 goods to Myra's Ocean Outlet
Part 3 of 4
Open 1 Turquoise Mine for operation
Make 2 Turquoise Opal Rings
Make 2 Turquoise-Button Pouches
Note: Here again, in Part 3, we have a handy overlap with Tanya's Preppin for Booty quest (also Part 3), as it is turquoise-heavy as well. However, the only actual shared element (owning a turquoise mine) need not be completed concurrently in order to count, so don't sweat it if you're not in the same place at the same time.
Part 4 of 4
Get 6 Goat Milk
Make 3 Herbed Goat Cheese
Sell 3 goods to Myra's Ocean Outlet
Note: If you're like me, you'll get this quest before you've received Tanya's Udderly Cute quest. While there are no direct overlaps between the two, you may want to put off starting this part until you've received the other quest. Reason being, if you get your goats and make your herbed goat cheese, you will, by default, be fulfilling two parts of her quest as well. However, if you accomplish these things before accepting her quest, you'll have to repeat the effort of making the cheese.