As Baurus told you after the Emperor's death, Jauffre can be found at Weynon Priory. It's one of the few times that you're given a quest and find an active marker on your map. Most of the time you get the marker, but you can't fast-travel there. This time you can.
Once you arrive at the Priory and ask around, you'll be told that you can find Jauffre in the priory house itself. Either Prior Maborel or Brother Piner will stop you and ask your business when you enter (it depends on which one is closest, but Prior Maborel will do the speaking if both are present). If you explain that you need to see Jauffre, they will direct you to where you need to go and be willing to help you after you speak to Jauffre. If you tell him that your business is private, you'll get the "oh, more strangers showing up in the middle of the night" routine and be told to go find whatever it is that you're looking for.
If you arrive during the day, Jauffre will be up the stairs to the right, sitting behind a desk. If you arrive at night, he'll be up in the sleeping area to the left. Just walk up and initiate the conversation. The Amulet of Kings will be removed from your inventory, the quest will be marked complete and you'll be given a new quest to find Martin at the Chapel in Kvatch. Jauffre will offer you assistance and if you pursue that topic he will terminate the conversation, get up and unlock the chest next to the bookcase. For that reason, I'd recommend saving the "Assistance" topic for last rather than have to reinitiate the conversation.
When Jauffre unlocks the chest, you'll find two sets of armor: a partial set of Iron (including those boots that I told you we'd pick up later) and a partial set of Leather, a bow and arrows, a few lower quality weapons, a few potions and a couple or three scrolls. Help yourself.
If you speak to Prior Maborel about assistance, he'll give you his horse (personally, the only advantage to a horse that I've found is that you can use the roads without having to stop for bandits - your mileage may vary). Brother Piner will offer you some insights into the more spiritual side of the Blades and a copy of "The Warp in the West," which will boost your Block skill when you read it. Eronor (the Dark Elf shepherd) will give you a Repair Hammer. There isn't much else to do around here, so unless you're into the bucolic lifestyle, it's time to move on.
For future reference, the beds here are available if you'd rather not fork out for an inn the next time you need to level up.