After crossing the bridge in Saria Town, Link can go into the cave that leads to Death Mountain.
On the other side of the cave, you will find yourself in a network of caves. First I will describe how to get to the destination, and then after this paragraph I will explain how to defeat the enemies in the caves. To get to the correct destination, whenever you have multiple choices of the cave to go into, always chose the one on the right. Eventually, you will exit from a cave and be in a desert. When you get to that desert, you should go to the west and then up and into the hole in the ground next to the boulder.
But before you can get to that point, you will have to get through the many cave action screens. Remember: when you are in the overworld and have multiple choices for which cave to go into, go into the one on the right.
From the Death Mountain entrance, go right and into the cave. This first cave has some Bots in it, as well as some spear Moblins. For the Moblins, just watch to see if it is going to stab high or low, and crouch or stand as needed to block the attacks. Then just stab the Moblin when you can until it dies.
In the next room, there are Octoroks on each step. Wait for them to spit their rocks, then kill one and move onto the next. Beyond these Octoroks, you will find a Blue Goriya. Carefully avoid its boomerangs while attacking when you can. Just watch for where the Goriya is holding the boomerang so that you know whether to crouch or stand in order to deflect it. And be sure to keep a little bit of distance between you and it, because if you are too close, then if he throws a really low or really high boomerang, it could hit you. Beyond the Blue Goriya are some more Octoroks. They are slightly more difficult to get rid of because of their positioning, but just wait for them to spit their rocks, then carefully drop down to stand in front of the nearest one, and kill it before it can spit another rock. Now you can exit this cave.
In the overworld, head to the cave on the right. In this next cave area, there are Bots and Aches that are pretty easy to deal with. Go past the elevator, because it takes you the wrong way. Soon you will encounter a Daira. It keeps swinging its axe at you over and over. My strategy for dealing with these things is to get on the same level as it, then crouch and stab as it approaches. It should get knocked back, allowing you to stab again while still crouching. You should be able to kill the Daira this way without getting hurt unless your timing is off. Don't try to attack the Daira if you are not standing on the same level as it. Just go back to the previous room and come back in order to respawn the Daira. Or you can just attempt to jump over it, although this is risky.
Once you are past the Daira and outside, go into the cave on the right, making sure not to linger in the forest where random enemies could get to you. Inside, there is a bridge over lava where you will have to deal with some Bago-Bagos. Fortunately, there are no gaps in this bridge. To avoid getting hit here, if you see a Bago-Bago flying up from in front of you, crouch down for a moment. This will protect you if the Bago-Bago spits a rock on its way up. If a Bago-Bago jumps up from behind you, try to either jump over any rocks it spits or just turn quickly and crouch to block it. If necessary, turn around and kill the Bago-Bago to keep it from running into you.
Once you are past this area, exit and go into the cave on the right. Cast your Jump spell here. Deal with the Bots in the way, and then jump up onto the first of the high ledges floating in the middle of the room. There is a Red Daira down below that throws axes. It is pretty difficult to kill. You have to watch to see if it is about to throw an axe, then jump to avoid the axe, and then somehow manage to hit the Daira when it's not throwing axes at you. If you are on the floating ledges, however, you can try to completely avoid the Daira, and you will be rewarded with a red jar. When you are on the third floating platform, try jumping at the very last moment in order to jump over to the fourth platform. If you don't make it, then you'll have to make a run for it to the right side of the room while jumping axes and avoiding Bots.
You will be in the overworld with only one cave to go into, so go in. In this cave, you'll have to fight some Aches, a blue Goriya, and an orange Daira. When you're past them, go outside and into the next cave.
This room has some Bits, which are red versions of Bots that can't jump, and there are orange Dairas in here, so don't hesitate to drop down to the same floor level, because otherwise they will get stuck on the step below you and you will have to have really good timing to try to stab them in the face without getting hit with the axe. Once you get past these guys, exit and go to the next cave outside.
In this next cave there are some Bots (the blue ones that can jump) and there are more orange Dairas. The stairs go up in here, so the Dairas can't get stuck on a step and constantly swing the axe like before. Defeat these Dairas and go outside.
You're almost done! You are in the desert, so go west and up through the narrow passage. There is a hole in the ground here. Jump in.
This area is very tough. This first room has red Aches, Lowders, and red Dairas. If you have enough magic, you can cast the Jump spell and try to avoid these monsters entirely.
In the next room, some Megmats will bounce toward you. After you get past them, there is an orange Daira.
Go down the elevator when you get to it. Don't bother going past the elevator, because that way is a dead end. After going down the elevator, go to the right (it's a dead end to the left.) Kill the red Daira or just use Jump to avoid it.
This room has a bunch of dangerous lava pits. There is even a Myu on one of the ledges. Link can't kill Myus at this point because they are too close to the ground for Link's sword to reach. Because of this, you should wait for the Myu to wander off of the edge of the platform before you try to jump there.
After you get past the lava area, you will encounter an orange Daira and a red Daira. Beyond the red Daira is the treasure of Death Mountain: the hammer! With this item, you can press A on the overworld to destroy boulders. This will let you get to the Moruge Swamp/Saria Town area without having to go through those annoying caves anymore! Just go south from Rauru Town and destroy the boulder, follow the dirt path, and you're there.
If you get the hammer but then you die soon after, it's not so bad, because you will always have the hammer in the saved game thereafter. You will have to come back to Death Mountain if you want the Magic Container that is hidden there, but when you have the hammer, you won't have to go back through the cave maze to get it. Instead, you can break the boulder near Rauru Town, head south along the path, and continue along the path until you reach another boulder in the path, this time on the right. (I don't mean the boulder that blocks a cave entrance near Rauru town. The boulder that I am talking about is farther south.) Break it, then head south quickly through the desert, follow the path that goes around the northwest part of the graveyard, cross the bridge, fighting Lowders and Moblins to get across, then enter the cave, go south until you reach a small cave with some lava pits, and then when you are back outside, head west through the desert and return to the place where you got the hammer.
To get the Magic Container, just break the boulder next to the cave opening where you got the hammer. Walk onto the spot where the boulder was, and you will fall into a pit. There is a Magic Container here. It will permanently add a segment to your magic meter.
There is nothing to do in the desert here, although you might notice that it is modeled on Hyrule from the original Legend of Zelda.
Now that you have the Magic Container from Death Mountain, you can exit this area. Go south to the desert, then all the way east to the water. From there, go north until you reach a cave. There are some lava pits and Red Octoroks to get past.
After getting through that cave, go north to get to another cave. This cave has no action screens. You end up in a small desert with a bridge. Go onto the bridge. There are red and blue Moblins and a Lowder to fight in the first screen, and then some poisonous bubbles and a red Daira on the second screen.
After getting across the bridge, you can go south along the coast to find an action screen with a 1-Up Link Doll in it. However, the doll will disappear forever if you take it, so you should probably save it until later in the game.
To the east, you will see a cemetery. Don't walk in the graveyard area unless you want to fight some tough random monsters. Stay on the path west and northwest of the graveyard to get past safely.
Head north across the desert from the graveyard, then head east along the path to get to Mido Town.