After the cutscene, Gulley is heard knocking at the door. Press to get through the dialogue. Once Gulley exits, use to follow him down and leave the house. Gulley points out the weather vane, which you should now use to save and mark it on your map. Head west a screen, then take the upper path after Gulley, then west again. Enter the house here. After some dialogue, you must pick up the sword by pressing . You're just delivering it, so you cannot use this sword at the moment. Exit the house.
If you look at the bottom screen, you'll see a red X marking your destination. Head north a screen then east to reach the castle. Speak to the guard at the gate, who will inform you of the captain's current location: Sanctuary. The X on the map has moved up, so head back west a screen, then north, then east to get where you need to be. Approach the two characters north near the Sanctuary. After the scene, you can swing the sword with B. Go east, and you should notice a grave with two bushes in front of it. Swing your sword to remove the bushes, then push the grave with to enter Sanctuary's secret entrance.
Your upward progression is blocked. There are actually pots in your path hidden by the darkness, so just swing your sword to get rid of them and move up to the chest. Press to open it and get the lamp. This will allow you to light torches, and gives off a bit of light around you in dark areas making it easier to see. To open the upper door, light the torches on both sides of the room, then enter after it opens.
Watch out for rats in the next room. You can light torches in your path to help you see. To the left is a chest with a red rupee, which is worth 20 rupees. The left stairs up lead to a locked door. Bypass it, following the upper grating path to the right to find a chest with the small key you need. Return to the locked door, press to open it and continue.
Once you enter the next room, you'll be trapped with three Poes. Just swing your sword at them to defeat them. Once they're gone, the upper and lower paths will be open. Head up to the next room. There are levers on either side. Ignore the right one since it just causes Rope enemies to appear. The left one'll allow access to the upper room. Continue on in the upper room to trigger a scene.
You will reappear in Link's house after the scene, and be introduced to Ravio. Ravio will give you Ravio's Bracelet.
Note: at this time there are 2 Piece of Heart available, but you won't be able to get a full container yet and won't come across any more Pieces of Heart until after the first true dungeon.
Head north back to Hyrule Castle. Talk to the soldier by the door, and Impa will eventually allow you to enter the castle, so do so. You'll have to wait a little bit until Impa returns, telling you you can see Princess Zelda. In the meantime, you can view the five paintings, which will give explanations upon examination.
After Impa says you can enter, climb the stairs on either side and enter the upper door. In the scene here, you will be told to visit Sahasrahla in Kakariko Village, which is indicated on the map. You'll also get a special charm from Zelda. Now, head down and while still on the upper level, head right a screen. Remain on the upper level and exit south. Follow the path along the wall to reach a chest with a purple rupee, worth 50. You can leap off the nearby ledge. Then, travel west towards the X. A music change indicates that you are now in Kakariko Village.
Enter the house indicated with an X to speak with Sahasrahla. Now, you must seek Osfala near the Eastern Palace, marked on your map. Before continuing on, there are some optional things to do in the village. First, head northwest and you'll see a cave, inside of which is a chest with 20 rupees. You can't get the heart piece here right now, though. There are some useful items to buy in Kakariko. To get everything, you'll need 150 rupees total. If you're lacking the necessary funds, a minigame can be found south of the village.
Items in Kakariko
To the northwest you'll see what looks like a house with a large shield atop it. This is the Item Shop, as indicated by the sign. You'll see some items, but what's most useful at the moment is the Shield, which you can buy for 50 rupees. With that, press R to ready your shield.
Just east of the item shop you'll find a merchant. Stand in front of the Bottle and press A to purchase it for 100 rupees. It'll go into your inventory. To collect certain things to hold in it, though, you'll need a net to catch them, so see below.
Bug Catching Net
Southeast of the central weather vane here, there is a house with a large bee sign. Inside, the Bee Guy will give you the Bug Catching Net.
To get this, head northeast to leave the town. Enter the hut on this screen to come across the fortuneteller, who gives you the Hint Glasses. Equipping these will allow you to get help from Hint Ghosts you come across.
Cucco Ranch Minigame
Find this minigame south of Kakariko. The objective is to avoid touching the cuccos by moving around. There are four difficulty levels, which determine the price to enter, allotted time, and number of cuccos. You'll start at the easiest, and upon passing each level, you'll unlock the next. In addition to winning rupees, your reward for completing the 'Rooster' level is a Piece of Heart. After that, you'll have access to 'Endless'. The number of rupees you get here is determined by how long you can go without getting hit. In addition, despite the level's name, you can end it by getting a time of 999.99, at which point you will be declared 'Cuccomaster' and win 3000 rupees.
To Eastern Palace
Once you've accomplished everything you want to do so far, head east towards the X on the map to progress with the game. Climb the steps up to the desert region, and you'll see pillars with bow icons, and the camera will move to point out two orbs. The signs here mention that Ravio now has a shop near the castle. Thus, head west back to your house. Inside, Ravio will ask which symbol you saw. Select the bow, which is the third one down. Then, Ravio will allow you to rent the bow for free.
Outside, you get an explanation of the energy gauge. Now, with the bow equipped, return to the place to the east with the bow icons and orbs. Stand on the platform with the steps. Shoot the orbs with the bow, and the upper path will open. Head straight for the X on the map, but be cautious of Octoroks and Armos. In particular, the two statues near the entrance of the palace will come to life when you get near and hop around, so hit them with the sword a few times to get rid of them.
At the top of the steps, you'll find Osfala. After mincing words, he'll enter the Eastern Palace, the first real dungeon of the game. You can save with the nearby vane, then follow him inside.