First, go here
http://wiki.ffxiclopedia.org/wiki/Category:Adoulin_Quests
There's a lot of quests you can complete in town just by walking and talking to npcs.
1 of them involve unlocking waypoints and is a good start.
Then do the missions here
http://wiki.ffxiclopedia.org/wiki/Category:Seekers_of_Adoulin_Missions
you'll be blocked at 1.5 very quickly and this is where you'll begin to go visit the adoulin zones.
Try to visit every zones you can and unlock waypoints.
You'll need certain key items to get access to some of the zones.
Work on them.
In this order, wiki thoses keywords and do the quests for them : Logging, demolishing, watercrafting, climbing. (there's a new one since November 4th update, i'm sure someone will tell us what it is.)
Now, you want to get your hands on level 106 gear. Get some baylds from reives (killing trees that block the way in ceizak battlegrounds).
If you have a good whm or a bard, you can skip this part and get into Skirmish for level 113 weapons and armor. (pre-adoulin geared DD's are not good in skirmish content. You'll miss 90% of the time and you'll die pretty quickly.)
BG wiki has good infos about Skirmish, read them.
With this gear you'll be able to participate in higher tier reives and delve fracture runs.
Hope it helped.