Head down the mountain and talk to Arngeir about the Elder Scrolls and he will point you to Winterhold in the north. Take the carriage from Whiterun to Winterhold and follow your marker to the bridge across to the College.
You will be stopped by a gatekeeper on the way to the College (the waypoint). Mention that you want to see Elder Scrolls and that you are the Dragonborn. Go inside and follow the waypoints to get to the library. Ask the mage there to see Elder Scrolls. Tell him you are the Dragonborn and he will get you two books. Take both of them and then ask him about the one that is illegible. He will tell you about Septimus and that he was out in the ice drift to the north.
Head outside but rather than go back across the bridge, head to the third window on the right. Get on the ledge, look out towards the river, and use the Whirlwind Sprint Shout. You should land in the river and not take a whole lot of damage. Now start heading north through the ice flows towards Septimus’ cave. When you get there, head down to him and he will tell you about how to get to an Elder Scroll. Don’t worry too much about all the things he says, you didn’t miss much. Once he’s given you the lexicon, fast travel back to Winterhold and start heading south to Alftand.
While travelling, I recommend that you try to stay as high up in the mountains as you can, it’ll always be easier to go down when you get close but be not to jump off of anything because there are a lot of spots to fall to your death. Once inside, you want to hug the left wall while fighting off mechanical enemies until you reach a room with a big light in the center. Take the north east door out of the room but watch out for a fire trap. In the next big room, you want to take the western exit and then take a left at the intersection.
Eventually you will exit out onto a ledge above a previous room. Be careful walking this ledge because there will be pistons that can push you off if you miss your timing. At the end of the ledge is a golden door that leads to a new section. Move forward through the next section, watch out for the Dwemer Sphere and use the oil on the ground to help take him down. Just beyond him is a set of stairs with pressure plates on either side of a rail leading up the middle. Try and walk on the rail so you don’t hit these pressure plates. If you hit one, quickly run down the ramp and then follow the spinning blades as they return to the top of the stairs. Open the gate with the lever on the floor and head down the twisting ramp beyond but watch for an ambush from behind. While on this ramp, make liberal use of your Unrelenting Force Shout to push enemies off the ramp to their death.
Eventually, this ramp will lead to a giant gold door but instead of going through it, you want to turn 180 degrees and jump of the edge to a lower platform. Off the side of this platform is another ramp platform heading down. Follow that ramp downward and you'll eventually come to big golden doors.
Walk through the room and down the stairs to find tents. Kill the inhabitants head into similar room beyond. This time there will be plenty of oil on the floor below you and the Falmer will present you with plenty of opportunities to maximize its effect. Work your way between the flame traps and you’ll come to a very difficult room.
I highly recommend that you save here because you will come up against varying Falmer magicians and they can be very difficult (didn’t you know that Falmer used to be Snow Elves?). Knowing the layout of this room can be very important. When you first walk in, go down the stairs and you can proceed almost straight ahead or go left. Going left provides room if you need it for fighting. If you take the straight path, you’ll find the path heads right and this will take you to an intersection. Directly ahead is an elevator that leads back to the entrance of Alftand while the left path leads to Falmer (which may have already seen you through the gated wall). This path leads where you want to go. Fight the Falmer using as much of the space as possible. If you have the spell Muffle, I highly recommend it here because Falmer are blind and this spell can give you the drop on them.If you can get beyond the Falmer, there will be a golden door and on the far side is a multilevel room. Bring the Falmer into here and you can shout them off of the ramp in this room. This won’t kill them so I recommend you try to only catch two of the three Falmer in your shout so that you can easily kill the third. Take the time from when you kill this enemy to back track farther away so you have more time before they get to you. Heal up and prepare your Thu’um for another push. Head back to the ramp, rinse, and repeat to kill the Falmer. This isn’t easy so have patience.
Once they are dispatched, head to the bottom but watch for the Frostbite spider, it would be terrible to die now after all that effort (Save!). Take the tunnel leading out of the room to the North and go through the big golden doors. Watch for the traps and go through the doors to Alftand Cathedral.
Go through this next room but watch out for pressure plates that will drop a trap on you. Follow the right path and watch for the Falmer here. Kill him and go through the door behind him.
Go through the door and you'll end up outside. At the top of the stairs will be two Falmer patrolling the area. Wait for a good opportunity for a sneak attack and then take them both out. Once they are out of the way, turn around and go up the stairs to the platform above. Hit the lever at the top and then head through the giant gate that was just opened. This is where it gets scary but if you keep your wits about you, it’s actually quite easy.
Head up the stairs and to the right once inside the gate. You’ll notice the giant mechanical golem come alive. You want to use the top platform, the bottom platform, and the two sets of stairs on either side to run circles around him. Let him come towards you and as he gets within range to use his frost breathe, head down the stairs and put the upper platform between the two of you. Go up the left stairs and shoot at him, then run down the stairs. Rinse and Repeat until you have a dead machine. Once you get the hang of it, it is possible to take him down (slowly) without taking any damage at all.
Go up the ramp here and you’ll come to a room with a couple of humans arguing over something. Ignore them and head to the mechanism in the middle of the room. Activate it and the floor will recess downward, revealing stairs. Go down the stairs and through the door at the bottom to find Blackreach.
Blackreach is a scary place and, for now, I recommend avoiding it entirely. There’s a lot of interesting things in here, like an old Dwemer city and crimson nirnroots which can only be found here. For now, head to the other side of Blackreach. Try to give everything here as wide a berth as possible and completely avoid the city in the middle. Once you get to the other side, you will find a tower with an elevator inside. Take the elevator up.
Go up the ramp here and then climb up to the control pedestal above. To the right side of all the buttons is a spot to put the lexicon. Put that in and then we start hitting buttons. The first two buttons move the sphere forward or back. Hit the left of the two buttons until the lexicon unfolds and reveals a blue center. Move to the left side of the center console and hit the third button until you see light shining down from the glass ceiling pieces onto the sphere. When that happens, the fourth button will open. Hit that to reveal the Elder Scroll. Go collect the Elder Scroll and then grab the lexicon. Leave through the doorway just under the control console. Use the elevator to get topside and then the lever by the left side of the gate to open it. Fast Travel to Septimus and give him the Lexicon. Travel Back to the Throat of the World.