The first thing you want to do is run around finding golden walnuts. First place to go is to the East, talk to the kid in the feather cloak (his name is Leo) and to the parrot in his tree house. You have to give your first walnut to this parrot to get access to the north part of the island. Once you find 10 walnuts total, Leo will warm up to you, start talking to you and you can start giving him gifts. (I recommend high quality duck feathers.) When it comes to walnuts, look for places where you have rocks or shells or plants arranged in a circle or cross or X shape and dig in the center with your hoe. Shake them off of bushes. (You also need to farm for some, break up rocks for some, fish for some, do a few quests, play darts with pirates. They're all over the place. There are 130 total.
When you go to spend walnuts, talk to the parrot by the big turtle first, get him out of your way so you have access to the farm space. Next unlock the dig site. These two get you access to the first areas you need to get a lot of stuff you need to do on the island. As soon as you can unlock the resort, that also unlocks the southeast area where you can find more walnuts.