The Second Quest is a much harder version of the game, where items are harder to find, some enemies are stronger and there are some new enemies, dungeons have different layouts, dungeons are more difficult to complete. This page explains how to start the Second Quest and what to do at the beginning.
How to Start the Second Quest
You can either win the game and press START after the credits roll to turn your saved game into a Second Quest slot, or you can create a new save slot with the name ZELDA to start the Second Quest right away.
Get the Wooden Sword
Go into the cave and get the wooden sword.
Now you should get some bombs. Blue Moblins and Blue Octoroks drop them, so go east, then north twice, then keep going east until you start encountering Blue Moblins. Explore this green forest until you have some bombs from killing Blue Moblins. You can carry up to 8.
Get 60 Rupees
There are some free rupees for the taking that will quickly get you to 60 rupees. You will need even more rupees soon, so get these rupees even if you already have 60 or are close to it.
If you are in the green forest as described in the "Get Bombs" section above, go to the northernmost area of the forest, where there are two armos statues. Touch the rightmost statue to make it move, then go down the stairs to get 30 rupees.
Go back outside. Go one screen north, then put a bomb on the wall to the right of the stairs. Go in to get 30 rupees.
Get the Blue Candle
From there, go south twice, then east, and touch the rightmost statue to reveal stairs. Go down to get 10 rupees. (You'll need them soon.) Then go west, south, and east. Go into the cave and buy the Blue Candle.
Get More Rupees
You are going to need 250 rupees. Go west three times and use the Blue Candle to burn the bottom-left bush. Go down to get 10 rupees.
Go west five times and burn the bottom-left bush to get another 10 rupees.
Then go west, south, west two times, and north. Burn the bottom-most bush to get 100 rupees.
Go west twice and burn the bottom-right bush to get 10 rupees.
Go east four times, then south, then east four times, then north twice. Burn the upper-right bush in the corner to get 30 rupees.
Go north twice and burn the bottom-right bush to get 30 rupees.
You should have a lot of rupees by now. If you haven't yet reached 250, go east and kill some red Leevers for a chance to get blue rupees, then go to the south-middle and southeast parts of the map to kill blue Tektites and red Leevers, both of which have a chance of dropping blue rupees.
Get the Blue Ring
Go to the southeast corner of the map, then go north until you can't go north anymore. Follow the water's edge until you reach the end of the water, and go east. You will see some trees. There is a secret passage north of the bottom-middle tree stump. Go up through the north wall there to find a hidden building. Go inside and get the Blue Ring.
Go to Dungeon 1
Now you should go to the big lake in the center of the map. On the east side of that lake, there is a bridge. Cross it and enter the tree to go to dungeon 1.