I was pretty sick of quests by level 40 or so. I felt that I was spending too much time traveling from one zone to the next, picking up item X for NPC Y and taking it to somone on the other continent, and losing bag space along the way. After level 40, I simply stopped doing "pick up quests" and focused on specific quest lines that had rewards of interest (i.e., an item I wanted or an easy and fun path to XP points).
From 40 to 50 I only did quests for Zul'Farrak and other random things that I picked up along the way that didn't seem tiring. Carrot on a stick was one of the motivators.
From 50 to 54 or so I did the Nimby quests and ran Sunken Temple a few times to get feathers and gear.
From 55 to 57 I did the quests in Un'Goro crater and farmed mats for RotV. The Un'Goro quests have "chore potential", but since you're there and the quest givers are all packed together in the north central region of the zone, it ain't too bad.
From 55 to 60 I did the quests involved in the Onyxia quest line. Ran BRD and UBRS a lot. One of the perks of the new person caps in raids is that you get more experience per kill. It is easy to level up with a group of 5 in BRD or Strath.