Princess Zelda is safe in the Sanctuary. She tells you that the Master Sword is needed to defeat Agahnim, and that the elder Sahasrahla will know more.
Before leaving the sanctuary, open the chest behind the priest and get a full heart container. You'll only be able to get pieces of hearts after this point, except for defeating boss monsters.
From the Sanctuary, head west and south to Kakariko Village. Press the map button and you should see the elder's house marked in the village. Along the way, remember to defeat as many enemies and cut as many bushes and tall grass as you can to get treasure. (You should do this for the entire game.)
In the elder's house, the old lady will tell you a little more of the Master Sword and to look for someone who will know where the elder is. Talk to as many people in town as you can, but be careful - Agahnim has claimed you kidnapped Zelda, and some people will call for soldiers if you talk to them. The boy at the chicken farm knows where the elder is - he'll mark the spot on your map.
There are still a few things you must do before leaving town. If you entered the tavern, you met some people with information. Enter from the rear and open the chest to get a bottle. You can get a second bottle from the merchant near the weathercock for 100 rupees. Now go to the cave in the northwest end of town. Don't enter the cave, but jump down the hole next to it from the hill above. Get the bombs from the chest, then plant one where the north cave wall looks different to get to a chest with a heart piece. Then go to the thieves' guild near the elder's house.
In the basement, get the rupees in the chests then bomb the crack in the wall to get another heart piece. Get the bug net from the sick boy whose house is near the center of town. Leave town the way you came in and head north to the Lost Woods entrance. Take the north path until you get to the thieves' hideout. Cut don the bushes behind the entrance to find the secret hole to a heart piece. Head north from the hideout and go through the left log to find the mushroom.
Got all that? Now we're ready to head for the elder near the East Palace.
Head back toward the sanctuary and this time take the road east and south. After you cross the bridge, go north first to the witches' hut. Give the witch outside the mushroom, then enter the hut and buy as much Red Potion as you can. Now go south of the bridge and follow your map's path to the elder. Sahasrahla will tell you to find the three pendants, and the first one is in the east palace. Before leaving the temple, bomb the north wall to find treasure behind it. Then follow the map to the East Palace. Watch out for the Armos Statues that will come to life on the way.
Once you enter the East Palace, take the middle of the three doors, using the switch under the pot to open it, then step on the risen step on the bridge to open the next door. Watch out for the steel balls that will roll your way down the next hallway, especially the big ones, and go up the steps to the next door. You'll have doors on each side of you that open by the switches behind the pots. First go east, take the upper door in the next room, hit the switch under the center pot to pen the next door and get the map. Then go back to the west door, take the next door at the bottom, defeat the Stalfos to go through the next room and get the compass.
Go through the door at the bottom of the stairs and touch the tile in the next room to hear from Sahasrahla. Take the door to your right and you'll enter the room with the big chest, but you can't open it yet. Take the top right hand door, go under the overpass and enter the dark room. Hit the raised tile near the right door to open it to the next room, and lift up all the pots until you find the key.
You can now open the door on the left side of the first dark room, then follow the path to the room with multiple enemies. Defeat the Stalfos, Leevers and the Cyclops, who can only be hit when his eye is open. When all those monsters are gone and the evil Fairies break away, hit the switch under the pot they were guarding to reveal the chest with the big key.
You can now return to the big chest and get the bow. You'll also get 10 arrows, and can find more under some pots now. See the two pots near the big door? You can jump down into them from the platform above to get the fairies inside; be sure to grab one with the bug net and keep it in a bottle; it will revive you if you lose all heart. Exit the pot through the gold portal and enter the big door. You'll need to defeat all the monsters in the big dark room to get the key, then use it to open the door to your left and go up the stairs.
In the first room on the second floor, lift all the pots to get more arrows as well as the switch. In the next room, the raised tile on the bottom will open the door. Watch out for the steel balls in the next room and hit the northwest tile to open the doors. You must defeat all the monsters in the next two rooms to move forward, but the red Cyclops here can only be hit with arrows with their eyes open. The Armos Knights await you behind the next door.
The Armos Knights should be defeated with arrows. If you checked under every pot on your way here, you should have enough. Stay in one of the lower corners of the room while they march in a circle. When they line up and march toward you, fire three arrows at one to defeat it. Repeat this until one remains, which will then start leaping toward you. Shoot an arrow at it and then get out of the way. When all the Knights are gone, you'll get a whole Heart Container and the Pendant of Courage.