I think when you hit 15 you get a message about going back to your home city to talk to someone about the Wayfarers. There are also a couple other NPCs you can talk to to get started if you can't get back into your home city, check West Karana and Mountains of Rathe. Talk to them and just follow the [prompt text] to get started.
Once you talk to them, you can go out and get your adventure stone. Find one of the Wayfarer's camps in East Commonlands, Butcherblock Mountains, North Ro, South Ro, or Everfrost. Each camp has a person you can talk to about getting started in the Brotherhood. You should be able to tell which one they are because they don't have a special tag under their name in parenthesis. Hail them and they'll ask you some questions. Answer correctly and you'll get your very own adventurer's stone. [HINT: I've found all you really need to say when they start asking questions is "Farstone Magic". Then you get the stone and that's it]
Once you have the stone, you can have the Magus at each camp teleport you between the various Wayfarer camps. Just hail the Magus at a camp and they'll tell you where they can port you.
You can use the Augment Sealer thingie or what's commonly called "The Birdbath" (it looks like a birdbath!) located at each of the Way camps to add your adventurer's stone to a charm. If you don't have a charm, you can get one from the person labled (Adventure Merchant) at each adventure camp at the cost of a few adventure points. You might need to actually do a couple adventures before you have enough points to get a charm. You'll also be able to get some better items, spells, and augmentations from the Adventure Merchant once you start aquiring more adventure points.
In order to do an adventure, you need to get a group together. There can be no greater than 7 levels difference between members of your adventure group. Once you've got a group, the leader right-clicks on an adventure recruiter and requests a mission. Once you get one you like, accept the mission and head to the dungeon.
Every time you get 10 adventure points for a Way camp, go back to that camp and hail the person who you would talk to to get an adventurer's stone. They'll give you a bit of info and sometimes they add some stats to your adventure stone.
I think that's the basics. If you want more detail you can check out
this writeup.