The walk-through section won't cover everything you can do in the game--just the major places and the items and upgrades you'll need to finish. To find all of the heart pieces and treasure charts, see the appropriate sections of this guide.
You begin your quest on the aptly named Outset Island. After the conversation with your sister Aryll, head over to the house at which she points and see Grandma.
Enter the house and climb the ladder. Talk to Grandma to receive the hero's outfit, which, as everyone will tell you, looks too warm. Hope you think that's funny: You'll be hearing about it over and over again. Grandma will ask you to go find Aryll, so run back to the lookout and climb the ladder.
Aryll will give you a telescope. Equip it with either Z, Y, or X, and then use it to look at the mailbox, located just outside Grandma's house. Watch the mailman as he takes off and the subsequent cutscene involving the huge bird. It's time to live up to your new outfit and act like a hero.
First, though, explore Outset Island some more. The only thing you need to do is get a sword--head to Orca's house (it's the ground floor of the two-story house next to Grandma's) and follow his instructions. He'll teach you to fight, and after you finish his lessons he'll give you the hero's sword. Pay close attention when Orca teaches you the parry skill--this is by far the most effective fighting move in the game, at least until much later. Do a few of the Outset Island side quests (covered at the end of this section), if you'd like, and then head into the forest, high on the mountain above the village.
To get to the forest, follow the path behind the house near the lookout tower. You'll come to a row of trees blocking the path. Cut them down with your sword, then proceed. When you come to the broken bridge, just run and jump across the gap, then continue on into the forest. The path through the forest is fairly straightforward. Continue straight ahead until you meet your first enemy, a bokoblin--fight it and then keep moving. Look around for rupees, including some hidden in a fallen tree stump. Two more bokoblins will fall from the sky. Defeat them, and Tetra will fall from the tree.
Once Tetra is free, Aryll will be kidnapped. Tetra will agree to take you aboard her ship if you can find a shield. Return to Grandma's house and climb the ladder. Grandma will give you the hero's shield.
Before boarding the pirate ship, make sure you visit Beedle's shop, which is a little boat just offshore. Buy the bait bag, some all-purpose bait, and one or two hyoi pears. You may also want to finish exploring Outset Island for some rupees.
Talk to the man crouching in the grass near Orca's house to learn how to catch pigs. Catch all three and take them up the hill to the house behind Orca's, then throw them in the fenced-in area. The woman will give you rupees for every pig caught.
Cut the grass at the house near the lookout tower (and elsewhere) to find some rupees.
Equip and use a hyoi pear. You'll take control of a seagull and can fly around the island picking up those elevated rupees.
That's all you can do for now. Talk to Tetra to board the pirate ship.