So, I just decided to try DF2014 adventure mode. Everything's fine, I just don't really understand a thing about quests. Why don't I have a pointer towards the task on my compass when not quicktraveling?
Did you actually get a quest assigned to you via the "ask for duties pertaining to your service as a hearthperson" (may not be exact wording) option, or did you just inquire about troubles?
Whenever I ask for duties as a hearthperson (either from the site where I start as a hearthperson, or by killing enough things to get other sites to agree to let me be a hearthperson) the noble just responds "enjoy these times of peace" and doesn't give me anything to do. Has anyone actually ever gotten assigned duties, or does this not work yet?
If you inquire about troubles, and then follow up to inquire about specific troubles, you can then talk to the NPC again and ask for directions to the relevant site (usually the first option, but sometimes the second. Do not ask for the location of the relevant person - you want the location of the site). Then, if you pay attention to the name of the site, you can go into the "sites" section of you quest log, select this name (filtering is helpful here), and it will show you the location on the map. However, you haven't really been given a quest to kill whatever is there. The NPC you talked to will not care if you kill whatever they were complaining was so horrible there (they will just say "it was inevitable" if you tell them about it). However, it is still useful to talk to NPCs to get locations of things you might want to fight, even though they don't actually care if you fight said things.
Long story short, the old quest system from DF2012 is gone, and AFAIK it is not possible to get a "quest" in the new version. If anyone else knows how to get assigned a quest in any form, I am very interested to find out. It is my hope that if and when the "ask for duties pertaining to your service as a hearthperson" works, that something would be added to the "agreements" tab reflecting your assignment, but like I said I haven't been able to make this work yet. Even if the town is currently under occupation by goblins, they still say "enjoy these times of peace", so either it's bugged/not written yet, or I haven't become "worthy" of doing any quests yet.