No Keys Trick
While many dungeons have hidden keys that you must find to continue, there is a trick to bypass them. Once you acquire the "Fairy Spell" walk next to a door and use it. You will fly right through the door. Save up enough magic and you can get through an entire dungeon like this.
Avoid Harder Battles
Typically when you step on certain tiles on the world map, a battle will occur. To avoid these battles remain still when the monsters appear. Eventually one will come close to you and step on the tile. Approach them then and you will have a much easier battle to fight.
Quick Experience Points
In "Zelda II" you need experience points to level up your attack and magic power. This is done by defeating enemies and there is one area in particular that gives you a lot of experience points for little work. After you've learned the "Downthrust" attack and the "Fairy Spell," enter the Island Palace and "Downthrust" the statue of the red Iron Knuckle. If the Iron Knuckle does not appear leave the area by walking left, return to the world map, and try again. When he does appear defeat him and enter the palace. Kill all of the major enemies that appear in this short castle for an easy 275 experience points.
Hidden Heart Containers
The first of four hidden containers is just south of Parapa Palace in a small forest. The second one is in a cave just south of the town of Rauru. You will need the Hammer to destroy the boulder to get in. The third one is one an island near the Water Palace. Walk left 10 spaces, and up four to find the Heart Container. The final Heart Container is near the final palace. Go right on the shore and eventually you will find it.