From Ruto Town, go south and into the cave. This cave is rather lengthy. You will first encounter some Octoroks. After that, there is a Red Goriya. Later, there are some Lowders and another Red Goriya. In the next room, there are some Blue Aches, as well as a Red Ache. When the Red Ache flies down, it turns into a gargoyle-like monster that spits a fireball. Your shield can't deflect fire, so jump over it, or just try to kill the Ache before it spits fire.
In order to exit the cave, you have to cast the Jump spell to be able to jump over the high cliff at the end. Once you do, you can exit the cave. The marshy area to the south is Moruge Swamp. There is a lone patch of forest to the east of the cave exit, and if you go there, there is an action screen with a fairy in it. This fairy will disappear until you save the game or get a game over, so you might want to leave it for later. Jump at the fairy to get your health refilled.
NOTE: If you get a Game Over, or if you save the game, you will unfortunately have to start at North Palace and then go through this cave every time that you want to get back to Moruge Swamp and the areas of this region. You will eventually get an item that gives you a shortcut, but not for a while. If you keep dying over and over, be sure to spend some time leveling up. It makes a big difference.
One of the squares of swamp that is just southwest of the boulder in front of the cave entrance has a Link Doll. This provides a one time 1-Up. Be careful, though, because this doesn't permanently increase Link's lives or anything. And worst of all, once you take the Link Doll, it disappears forever! You should really leave it alone until near the end of the game.
You can head southeast from the cave until you reach a river and a dirt path. If you cross the bridge, you can get to Saria Town. However, the northern bridge is really dangerous, so I recommend taking an alternate route to get there.
Before going to Saria, though, there is a hidden cabin in the woods north of the bridge that you should visit first. There are a bunch of hidden action screens in the forest, but you can avoid them completely if you go this way:
In other words, stand on the dirt path above the fourth square of forest. Then go up onto the third square of northern forest. Then go left one square.
This is Bagu's cabin. He will give you a note to show to the River Man. This note is with you permanently, even though it doesn't show up in your inventory.
Now you can exit from Bagu's cabin and go south to the dirt path, making sure to avoid the action screens as indicated in the image above. Now you should go to Saria town. As mentioned before, the northern bridge here is really dangerous. If you do decide to use that bridge, then be sure to crouch if a Bago-Bago flies in front of you in case it spits a rock, and be sure not to hit a Bago-Bago when jumping over the gaps.
To avoid the Bago-Bago bridge, though, go east along the dirt path and south into the forest. Beyond the forest, there is a small area of desert with a single tree in it. This tree has an action screen with a fairy in it. Remember: if you use the fairy, she will not reappear until you save the game or get a game over.
Now you can head across the bridge west of the fairy action screen. This bridge has a bunch of poisonous bubbles floating up from below. These are easier to avoid than the Bago-Bagos in the north bridge. You will also encounter some Lowders, but these are easy to deal with. Make your way across, and you will be near a single patch of forest. In this forest, there is a P-Bag. It disappears forever if you take it, so only get it if you are sure that you will level up before you save or get a game over.
Head west and then south along the dirt path until you reach Saria Town. The man at the entrance warns you that the eyes of Ganon are everywhere. What this means is that some of the seemingly normal townsfolk here are actually spies for Ganon. Sometimes when you talk to a random person who is passing through the town, the person will say a normal thing but then turn into an Ache. This is a spy! You can go ahead and kill it.
To avoid these people altogether, just don't talk to any of the random people in town at all. There are only five non-random people in this town, all of whom are safe to talk to: 1) the guy in red near the entrance of town, 2) the woman in the purple dress who walks around in front of her house, 3) the plump woman in the purple dress who walks near the graveyard, 4) the magic-refilling lady, and 5) the life-refilling lady. Everyone else you meet outside could be a spy. None of the people indoors are spies.
Go to the second screen of town, and go into the door near where you come out. The house is empty, but if you attack under the table, you will find a mirror. Go back to the woman in the purple dress who wanders around in front of her house in the first screen of town. She lost her mirror, and you just found it, so she lets you into her house. Go to the back and talk to the guy in the basement to get the Life spell. This spell replenishes your life by a few squares. However, it costs a lot of magic, so be careful not to use it unless you are really low on life. This spell is particularly helpful if you find a red bottle. You can cast the life spell and refill your magic right away with the red bottle.
Go farther into town to find the magic-healing lady and the life-healing lady. In the house between their houses, there is a sleeping Bot. If you talk to it over and over, it will eventually explain about the hidden cabin in the woods, which you should have already gone to if you have been following the walkthrough. If not, check out the image farther up the page to find out how to get there.
For the next part of the walkthrough, I will talk about how to get through Death Mountain. However, you don't have to do it in this order. You can go to Midoro Palace (north of the swamp) first if you prefer. But if you are going to Death Mountain first, just keep reading.
Now you can go to the west part of town. There is water just beyond the small house and your jump spell does not improve your jump enough to allow you to jump across. Instead, go into the small house. If you have Bagu's letter (see above), you can talk to the guy in the house and he will create a bridge for you. Go across and exit the town.
You will be on the other side of the river, near the entrance to Death Mountain.