Start your search for walnuts at the island docks and work your way through each area systematically, it's easier than trying to bounce around. The volcano dungeon is probably your best first stop since you can grab about 17 walnuts from mining, fighting monsters and checking chests, plus there's two more in bushes near the entrance. While you're exploring the different regions, keep an eye out for that simon says puzzle in the tiger slime grove cave, and don't forget to check bushes and dig spots everywhere you go.

Getting those first easy walnuts will help build momentum. Talk to professor snail about fossil collections, and make sure you bring crops for the gourmand frog - they want specific things like melons and garlic. Fishing is worth doing early too since you can get up to 5 walnuts that way, and breaking mussel nodes around the island will net you another handful.

Something interesting - if you're farming on the island there's a chance crops will drop walnuts when harvested, and playing darts with pirates on even-numbered nights after 8 pm can get you 3 more. Journal scraps will point you to some hidden ones behind curved palms and near diamond shapes. If you get stuck, the parrot in leo's hut gives hints about ones you're missing, or there's always the golden joja parrot who'll sell hints for 10,000g each as a last resort but try to save that for when you're really stumped.