Weynon Priory and Cloudruler Temple
It doesn't really matter how you get to Weynon Priory with Martin (foot or fast-travel or a combination of both), as long as you get there. When you arrive, you'll see the brothers being attacked by Mythic Dawn assassins. There are five assassins: three in the exterior and two in the Chapel. The assassins seem a bit easier than the ones you had to fight with the Emperor, but that could be because you're better equipped by this time.
The assassin chasing Eronor will attack you and Martin will, of course, wade into the fray. Prior Maborel is history, Brother Piner is fighting two assassins and Jauffre is nowhere to be seen. If you help protect Brother Piner, he will go into the Chapel to help with the assassins in there.
After disposing of the three assassins outside, go into the priory itself and up to the library where you talked to Jauffre. You'll find that a previously locked door has been broken open and the room behind is a shambles. Check out all of the containers and then head for the Chapel.
You'll find Jauffre, Brother Piner (if he was still alive) and two or three Mythic Dawn agents inside. Once again, Martin will wade in. He doesn't seem to think that keeping the imperial backside out of harm's way should be a priority at this point. If you take out one agent, Jauffre, Martin and Brother Piner can handle the other. Actually Jauffre and Martin are both "essential" at this stage, so I don't think either CAN be killed, but I'd rather not chance it.
After the agents are occupying some other plane of the hereafter, Jauffre will want to go get the Amulet of Kings. If you checked out the priory as I suggested, you'll be able to tell him that it's gone and can move on to other things. If not, he'll head for the library and find out on his own.
Jauffre will then suggest getting Martin to Cloud Ruler Temple, which is north of Bruma. If you've explored out that way, you can just fast-travel there. You do get a map-marker, but I can't remember whether it's usable or not. If not, you'll have to travel to Bruma and go on foot the rest of the way. It's not far and there won't be much to fight along the way. If you'd rather go on foot, you're welcome to take that course, too. I never do, but it might be interesting to watch two "essential" NPCs fighting the stuff that you'll encounter on the way. At the very least, you can collect some more loot by taking that option.
If you give them time to do so, Martin and Jauffre will take horses from the stable (the third horse was Prior Maborel's; now that he's no longer capable of using it, it's yours if you want it). However you do it, you'll eventually be standing in front of the big gates of Cloud Ruler Temple, which will now open. Jauffre will introduce Martin as "Martin Septim" and he'll be hailed by the Blades and make a little speech.
Speak to Martin afterward to get a little background information on him. For example, he knows more than any priest normally would of Daedric magic and gates to Oblivion. These Oblivion Gates should be impossible. Portals to Oblivion, by their very nature, should be wildly unstable things. But in any event, he feels that he will be safe here and that you should go do whatever it is that Jauffre wants you to do.
Afterward, Jauffre will come offer you membership in the Blades if you do not have a bounty. It's up to you whether to accept or not. Aside from an Akaviri Katana, free armor and a place to sleep, there really are no benefits to joining the Blades. The faction has no quests and there will be no promotions. But from a role-playing perspective, you do get the reassurance of working for the good guys and there is all of that free armor and weaponry laying around. Thieves Guild members (or potential members) should go read the page on the "Independent Thievery" quest for a couple of good reasons not to join at this time.
Jauffre also wants you to go back to the Imperial City and find Baurus to see if you can get onto the trail of the Mythic Dawn. This is the start of the next quest, "The Path of Dawn".
After speaking with Martin and Jauffre, take a few minutes (real time) to watch the two Blades sparring on the practice mat. You'll get a 2-point boost to your Blade and Block skills for your trouble. You'll also find a couple of skill books inside the Temple, should you choose to explore before heading off into the wide world again.
If you'd care to hang around Cloud Ruler Temple for a while, you'll likely hear some interesting conversations between Martin and the Blades. He's not terribly comfortable being the "big chorrizo," but the Blades are having a hard time treating him as anything else. You might also note that his name has changed. Where he used to be "Brother Martin," he's now just "Martin." If Martin and Jauffre brought horses from the Priory, they'll be in the stables on the west side of the compound.
If you chose to join the Blades, you now have an Akaviri Katana. It's a decent weapon. It doesn't hit quite as hard as a Claymore, but it's one-handed, so you can carry a shield. You can also pick up a complete set of Blades armor from the west wing. The Cuirass will be in one of the upstairs rooms and the rest will be downstairs. It's unenchanted heavy armor, and provides a good compromise between weight and protection.