The Path of Dawn
Martin suspects that the Dragonfires are the key to closing the Oblivion Gates permanently. In order to do that, two things are necessary: the Amulet of Kings and the Emperor. You only have half of the key (Martin), so finding the second half is your first priority.
Jauffre sends you off to talk to Baurus, who stays at Luther Broad's Boarding House in the Elven Gardens district. He should be on the ground floor, sitting at the bar. There's a fellow reading over in the corner (Astav Wirich). You can talk to him if you'd like. Baurus, looking a bit different without his Blades Armor, will initiate a conversation with you if you sit down next to him and tell you that he's going to get up in a second and expects that Wirich will follow him. You're to follow Wirich.
He'll head down into the basement with Wirich right behind. About the time that Wirich gets to the bottom of the stairs, Baurus will lay into him and he'll trasform into one of those Mythic Dawn assassins. I don't think that Baurus is "essential" at this point. At a later point in this quest, he is definitely non-essential ("if he dies, he dies," as the devs commented), but I haven't seen anything in this quest that ever sets him as being essential in the first place, so do what you can to keep him alive. When you search Wirich's body, you'll find a copy of Commentaries on the Mysterium Xarxes 1, which will increase your Conjuration skill. After speaking with Baurus, he'll send you off to speak to Tar-Meena in the Mystic Archives at the Arcane University. If you've already done the "Necromancer's Moon" quest, you'll know who she is and where to find her. If you aren't a member of the Mages Guild, look for her in the lobby of the Arch-Mage's Tower.
Tar-Meena will tell you that the Mythic Dawn is a Daedric cult dedicated to Mehruns Dagon. Mankar Cameron's Commentaries is a set of four books. Volumes 1 and 2 are not too hard to find (she'll give you her copy of volume 2 - it increases your Destruction skill), but volumes 3 and 4 are a lot more difficult to get, but perhaps you could get more information from Phintias at First Edition Books.
There are two ways that you can approach this next part, cheaply or not-so-cheaply. If you speak to Phintias, he'll tell you that Volumes 3 and 4 of the Commentaries are extremely rare. So rare, in fact, that he's never even seen a copy of Volume 4. He does have a copy of Volume 3 that he's holding for another customer, though. If you can get his disposition up to 70 and he'll sell you the book for 100 gold (it raises your Illusion skill). Gwinas will come in shortly after the transaction and give Phintias a hard time for selling his book. You can button-hole him for information, but he's initially dismissive of whatever you have to say. When you tell him that the Mythic Dawn was responsible for the Emperor's assassination, he crumbles and tells you how the cult functions. He turns over a note from someone called "The Sponsor" that details where a meeting has been arranged so that he can get Volume 4 and you're on your way.
If you'd rather go the cheap route, you have two options. The first, and perhaps easiest, is simply to wait for Gwinas to show up to collect the book. You'll have to do a bit of intimidation to get him to cut loose with it, though. Just follow the conversation threads until you get to the part about the Mythic Dawn being responsible for the Emperor's assassination. You get the book and the note and are done.
The second cheap route is for those with a shadier bent. After you talk to Tar-Meena, a previously empty chest in First Edition now contains Volume 3. It's on the table next to the door to the basement. Don't talk to Phintias beforehand because his conversation about the book causes Gwinas to spawn and head for the shop, which will mess up the whole plan. But if you break in and steal it, though, well....you get the idea. After you have the book, you can talk to Phintias and get Gwinas' location at the Tiber Septim Hotel.
Phintias eats at the Merchants Inn in the Market District from 8pm to 12 am on Monday, Wednesday and Friday; and at the Tiber Septim on Tuesday and Thursday. He eats at the shop on the weekends. He sleeps from midnight to 6 every day.
After you have the book, you'll still need to get Gwinas' note giving the location of the meeting. You can visit him at the Tiber Septim Hotel and use the same intimidation tactics as your other two options. If you're even shadier (and very good) you might be able to just kill him and take the note from his body (he sleeps from 10pm to 6am and eats in the hotel) or you can pick his pocket. If you talk to Phintias about the books and then wait for Gwinas to come in and have his conversation with Phintias, you can tail Gwinas back to the Tiber Septim Hotel. He stops just inside the door and waits there for a while. If your skills are up to the task, you can just pickpocket the note from him. Otherwise, you'll have to break in to his room and either pickpocket or kill him while he's sleeping. Note that killing him will undoubtedly draw the attention of the Dark Brotherhood. Gwinas is non-essential once you have the book; in fact, he's removed from the game when this quest is finished and is, therefore, completely expendable.
No matter how you proceed, you should now have both Volume 3 of the Commentaries and the Note to Gwinas. If you're a thiefly-type, you're going to be sorely tempted to go check out the sewers on your own. Feel free to do so (you shouldn't run into anything heftier than rats, mud crabs and goblins), but be aware that a timer is running. If you don't go talk to Baurus soon, he will come looking for you, even in the sewers. Baurus is still "essential," so he can't be killed on the way, but he can certainly have his health knocked down to almost nothing. So when he decides to charge in on the critters in the sewers, be prepared to back him up. You might even consider running interference for him if your combat skills are high enough.
When you get to the proper location in the Sunken Sewers, Baurus will suggest that you split up (Baurus stops being essential at this point - "if he dies, he dies," as the devs commented in the script). He'll go meet with The Sponsor and you can cover him from above. This is your call. Baurus is the same level as you are, with approximately the same stats and skill levels, adjusted for being a Redguard. Aside from his Akaviri Katana, he doesn't have any magical backups or enhancements. If he goes for the meeting, you're going to have to provide the cover. If you insist on going for the meeting, you're relying on him to provide the cover and whoever is providing the cover is going to have to deal with two Mythic Dawn guards.
The meeting is going to end in a fight, regardless of how you go at it. The fight is going to start when the guards discover Baurus (or you) on the bridge. Or if you escape detection by the guards, The Sponsor is going to recognise either Baurus or you and denounce you as an imposter. Since I'm rather fond of Stealth-based characters, I typically take the bridge and try very hard to get in one really good backstab if the guards don't detect me. If they do detect me, I just wade into them and keep my fingers crossed on Baurus until I'm done upstairs. The problem with putting Baurus upstairs is that there isn't much of a way that you can help him from below aside from missile fire and ranged attacks or healing. From above, you at least have the option of dropping down if he's too hard-pressed.
The Sponsor (Raven Camoran) has the fourth and final volume of the commentaries on his body. If Baurus is still alive, he'll send you back off to Tar-Meena. If not, your journal will. At this point, too, Phintias has an endless supply of volumes 1-3 if you should need it.
You have a couple of chances to come up with the solution to locating the Mythic Dawn on your own. Otherwise, Tar-Meena will just hand it to you. When you first talk to her, she'll tell you to look for clues in the four volumes. If you come back a day or so later, she'll tell you to look at the first word in each paragraph. If that's not enough, come back a day or so later and she'll just give you the solution: "Green Emperor Way Where Tower Touches Midday Sun" (it's the first letter of each paragraph, starting at volume 1). That completes this quest and starts you into the next one, "Dagon Shrine". You do not have to go back and talk to Tar-Meena unless you want to. You can just proceed to the next step and the quest will be automatically closed when you start the next one.