As the owl is informing you, go straight ahead towards the Hyrule Castle.
Castle Town Market
On Link's right are located the medicine shop and the shop where you can find the Hylian Shield for 80 rupees, but you will be able to buy it later for 60 rupees or even get it for free! Go across the plaza towards the castle, then come back to the market and head back again to the castle (so Malon will appear a little further).
Note: To get a Piece of Heart, look for the chubby lady's dog.
Hyrule Castle
After the owl's speech, talk to Malon, the little girl. She will give you the Weird Egg. Then climb the vines next to her, go straight ahead, go down the ladder and get out of the small room. Follow the path, climb the hill on the left without getting noticed by the guards and go north-west. At the end of this path, there is a wall with rocks sticking out of it. Climb the vines here, climb down the hill and jump into the water. Turn right and swim to reach the corner in front of you on the right. There, get on the bank, go left near the sleeping man and give him the Chicken during daytime. Talk with him, then push the two blocks into the water, climb on them, jump to the other side and crawl beneath the door. If it is night-time, wait for daytime to get inside.
Go straight ahead while sneaking past the guards, then, when you see a window in the background, go towards it and turn left. Go across the garden and speak with Princess Zelda! She will tell you the Legend of the Triforce and confide her worries to you. Answer "yes" or "OK" to her questions and she will eventually give you Zelda's Letter. Upon leaving, you meet Impa, the princess's nurse belonging to the Sheikah tribe. She teaches you Zelda's Lullaby (you have to repeat it with the yellow buttons), and makes you exit the castle. After telling you about Death Mountain and the village where she was born, Kakariko Village, she vanishes. Head towards the south of the map and follow the path until you reach the Lon Lon Ranch.