For me personally, it all comes down to whether I care about the "World" I am questing in or not.
In WoW I never cared about the overall world or Storylines of the world so all the quests were boring, repetitive and just a means to Level Up. Pretty much every MMORPG I have played has been this way as well. The only exception was FFXI.
Although FFXI's quests were (maddeningly) vague and the rewards were usually a joke, I actually enjoyed doing most of them because I cared about the "World" in general and to some extent the NPC's that the quests revolved around.
This is what I hope ARR's quests are able to achieve. I don't mind killing X creatures for Y reward if I am somehow manipulated into caring about the NPC. If the NPC characters are interesting and possess some level of personality, then I will be more inclined to do their task for them and feel a sense of accomplishment.
But to do this requires a very good writing team who takes the time to intertwine the NPC's stories with each other. I am not necessarily a good writer myself but I will try to give an example:
It's not enough to just have an NPC say, "Oh my child has been snatched by some imps, please rescue her." That is boring.
However, if NPC Tonah says, "I run an orphanage for the children who's parents lost their lives in the Garlean Battle and the destruction caused by Bahamut but we are always in need of supplies of food and clothing. (NPC) Shayla is a Leathercrafter who usually comes once a week with her daughter to donate leather for clothing and crafts for the children. Her daughter is so sweet and quite the little crafter herself. She always sits with the children and shows them how to make crafts with the leather scraps. They always look forward to Shayla's visits but we have not seen them for weeks now and I am worried what has happened. Would you please check on Leathercrafter Shayla and her daughter and see what is the matter? She can usually be found at XX,YY Location in Gridania."
You travel to XX,YY Location and learn from another NPC that "Shayla and her daughter went out hunting Hoglet Boars for their skins. Her daughter, Camille was so excited to go on her 1st hunt but I feel something terrible has happened. They said they were heading towards XX,YY as the boars there are not too large for Camille to handle. Please check on them?"
You travel to XX,YY and find Shayla sobbing because "we were attacked by Imps and Camille was carried off. Please I cannot return home without her. The Imps headed off towards XX,YY. Please find her and bring her back to me"
You find the Imps, kill them, rescue Camille, return to Shayla. She gives you the Boar Skins that they had collected to take to the Orphanage and teaches you how to make them into Boarskin Leather.
You take the Boar Skins and either craft them yourself into Leather or find a Leatherworker to do it, then give the Boarskin Leather to the Orphanage. Tonah is so grateful that Shayla and Camille are ok she offers you a piece of gear that the children have made. One of the orphans is so grateful that Camille is alright that he offers you his fathers Weapon and hopes you can help many more of Eorzea's children with it.
I do not expect every quest to be this involved but if at least some of them are, I will be very happy.