The Elder Scrolls V: Skyrim Special Edition

Main Quest, therefore, unmissable. Obvious spoilers below.

There are two ways to start this quest:

1) Speak with Arngeir at High Hrothgar
2) Speak with Esbern at the Sky Haven Temple

You'll be instructed to head to the College of Winterhold to learn about the Elder Scrolls (if you have not yet visited Winterhold, visit any stable in any major city to use the carriage to get there quickly).

You'll need to prove you are worthy of entering the College once you arrive, you can do this by casting a spell, using a shout, or persuading Faralda.

Once you gain admittance, seek out Urag gro-Shub in the Arcanaeum. He'll, eventually, grant you access to a few books detailing the Elder Scrolls, and the location of a man who gazed upon one of the Elder Scrolls.

EDIT: You CAN head straight to Septimus' last known location (Septimus Signus' Outpost) and avoid the College altogether)

Head north of Winterhold to locate Septimus Signus' Outpost, you should be able to see it in the distance due to the torch that illuminates the surroundings (if you're skipping the College, and therefore not getting the objective to locate Septimus, try doing this at night, as it should easily be visible).

After finding Septimus, he'll inform you there's a way to read the Scroll without losing your flippin' mind, as well as the location of the Scroll, he'll give you the Attunment Sphere, and Lexicon.

Head to the ruins of Alftand and Blackreach, these ruins are filled with ancient Dwarven tech (stock up on scrap metal, plate metal, and struts if you are interested in making Dwarven armor or weapons, these can be smelted to make Dwarven Ingots) and Falmer. It's a pretty long fort, made up of ruins and an underground passage.

Once out of Blackreach, you'll be in the Tower of Mzark, where the Scroll resides. You'll end up in an observatory like room, go up the ramp and place the Lexicon in the pedestal and hit the taller of the two buttons until the bttons to the left become active. Hit the new, taller one until the one on the left is active, hit that and you'll be presented with the Scroll.

Pick up the Scroll to complete "Elder Knowledge", and the Lexicon to complete "Discerning the Transmundane"