After beating the swamp, your next target is the mountains. You absolutely need bombs (or the Blast Mask) to scale them, so if you don't have any of these yet, return to the Bomb Shop and get a Bomb Bag and/or bombs.
The mountains are found to the north of Clock Town. As you make your way there, your progress is quickly stopped by ice - but don't fret, look above and you'll notice an icicle. Hit it with two arrows and it'll fall down and break the ice barrier. The path up is pretty straightforward, but enemies like to hide in the snow so be careful. Around the middle, your progress will be blocked by large snowballs - blast them away with a bomb and continue scaling the mountains.
Mountain Village is where the Gorons live; however the whole area is covered under a thick blanket of snow right now. If you go forward you'll find an Owl Statue to activate. The path to the north is impassable due to large gaps, so continue to the east. In the next area, cross the bridges and you'll come across Tingle, who'll sell you a map of Snowhead for 20 Rupees. Once that's done, continue to the next area.
You're now in Goron Village. You can go inside by speaking to the gatekeeper (on the first day only, on later days he'll be frozen solid), but there isn't much to do in there right now. Instead go to the opposite side where you see a floating island in the distance. You'll meet the owl, who'll fly over and drop some feathers on the way. These land on invisible platforms, showing you how to get across - hurry because the feathers don't stay forever. Once you've made it across, enter the cave and open the chest there to get the Lens of Truth. You can try it out right here by discovering an invisible chest containing 20 Rupees (guarded by an invisible Skultulla). Blowing up the rocks reveals another chest with 50 rupees.
Get out and this time use the Lens of Truth to reveal the platforms. On the other side you'll come across a dead Goron, who is invisible to you without the Lens. Agree to follow him, and he will lead you all the way back to Mountain Village, to the large stone wall. With the Lens of Truth, you can see there are invisible ladders all across the wall, allowing you to scale it. Once at the top, follow the Goron into the cave. Inside, you'll find out his name's Darmani, and he died while going up to Snowhead Temple. Once he's done, play the Song of Healing to heal his soul, which will net you the Goron Mask!
If you want, read the grave for some instructions. Afterwards, pull it back to reveal a hot spring. Get out your bottle and catch some Hot Spring Water. It won't stay hot forever so hurry out and run back to the bridge area. Stand next to the ice blocking the hole and use the Hot Spring Water to melt it, which will reveal a hole leading to a second hot spring. Anyway, get back to Goron Village, use your new Goron powers to pound the ground next to the door ( + ) which will open the door, then go inside.
There isn't actually much to do in there right now. The shop is at the bottom, but it's horribly overpriced. What you'll want to do is go up and follow the red carpet until you reach the elder's son. Talk to him as a Goron to find out that he misses his dad. Afterwards, get out.
Go out and find the elder, who is frozen in a huge snowball. If it's the Final Day you'll probably find him next to the Owl Statue, otherwise look around in the bridge area. Once you know where he is, know that you'll need Hot Spring Water to melt him - so go to the nearest spring, catch some Hot Spring Water, and then run back to the snowball. Destroy it as a Goron, and use the Hot Spring Water to melt the elder. Talk to him a few times and he will be sad that his son misses him. He wants to teach you the Lullaby, but because he can't remember it all, he will only teach you the beginning:
Now, get back to Goron Shrine and play the Lullaby Intro to the elder's son. He will teach you the complete song:
He will then promptly fall asleep.
There's now a sidequest you can do. Now that the Gorons are asleep, the torches will be lit. Pull out a Deku Stick, go out and light all the other torches in Goron Shrine. Once you're done, the large chandelier at the top will start spinning. Get to the carpet, transform into a Goron, and roll up to destroy the jars. One of the jars contains a piece of meat, which will fall down once that jar is destroyed. This can only be carried by a Goron, so pick it up and walk back all the way to Mountain Village. In a corner you'll see a Goron who is starving, throw the meat up to him and he will give you Don Gero's Mask.
In Goron Village, if you have the Big Bomb Bag, speak to the Deku Scrub as a Goron. He will offer you the Biggest Bomb Bag in exchange for the Big Bomb Bag and 200 Rupees. Afterwards, return to Mountain Village, and this time, follow the north path past the Owl Statue.
In here, simply curl up and roll across the gaps. Be sure to hit the snowballs on the way to get some much needed magic. At the end you'll reach Snowhead.
You'll end up right next to an Owl Statue, so activate it. A big gust makes progressing impossible - to stop it, transform into a Goron and play the Goron Lullaby. As it turns out, a Goron has been blowing the gust all the time, and as he now falls asleep and slips off, the path is free for you to traverse. Still, because it is so narrow, be careful to not fall down. Once you're at the top, roll around clockwise to reach a ramp that leads right up to Snowhead Temple.