The quests start in Mulgore. Skorn Whitecloud, a grizzled old Tauren Hunter, tells you a story about a legendary wolf named Ghost Howl, a noble beast who went insane due to an injury suffered at the hands of a demon in the old wars. He doesn't give you a quest to kill the wolf, he just relates a story of his glory days and a legend of his people. It's a simple act that makes you feel immersed in the world, without the nagging feeling in the back of your head, that voice that says, "man, I have to go track down this dude and kill him now, don't I?"
Yet, as you run around the plains of Mulgore taking care of other quests, you may have your encounter with Ghost Howl himself, and when he dies, you'll find his hide, which you can bring back to Skorn for his approval. It feels less like a chore of a quest, and more like a brave and noble hunt you undertook by yourself.
This basic pattern continues into the Barrens. You can defeat Lakota'mani the kodo, Washte Pawne the tunderhawk, and Owatanka the thunder lizard, bringing trophies of your hunt back to the Tauren, taking part in the ritual of the hunt and being hailed as a great hunter and a friend of the spirits. You can also kill the strange Harvester, ushering in the first signs of a sinister infestation that will grip all of Kalimdor, an epic storyline that comes to fruition nearly 45 levels later in Silithus. In a more non-hunting vein, you can also assassinate an Alliance messenger and deliver her note to your superiors, leading to a quest to "send a message" to the Alliance.
In Ashenvale, Senani Thunderheart will advise you on the great hunt, but it is your choice to go out, and using her directions, find and kill your prey, bear, cat, and hippogryph. At the end, as a recognition of your mastery of the hunter, you'll receive accolades and a cloak.
Now, Drysc has rightly pointed out that Blizzard's gotten better and better at implementing varied and interesting quests, and I think their quest system has a lot going for it. But there's also room for improvement. For me, one of the easiest ways to improve would be to go back to the beginning and use those Barrens quests just a little bit more often. Let picking up quests feel a bit more organic. Rumors in town lead to you watching out for certain monsters or landmarks or roving bands of thugs out in the field without being specifically on a quest. From there, you pick up the quests in the field.
Syp also wrote about a similar idea on the Waaagh blog a few days ago, and I agree: this type of dynamic questing would serve to make playing a lot more exciting and make people feel like their characters are so much more involved in the story and world. Being able to take quests on split paths would be great too. Say you intercept a messenger from the Venture Company speaking about a stealth clear cutting operation in Ashenvale. Do you send word to Astranaar and help them ambush the logging crew, saving the trees and becoming a hero of the Night Elves? Do you negotiate with the Venture Company to keep quiet about the operation for a cut of the profits? Or do you offer to help them slip past the Night Elves and into Ashenvale, setting up the possibility of further business dealings with them on down the road?
The biggest problem with this, I suppose, is that people who aren't natural born explorers might feel a bit left out without that Carrot on a Stick to lead them out to the frontier to find all these quests. Of course, with that, you can just give them quests at towns and quest hubs that will entice them to head out and pick up the "real" quests in the field. However, keep fewer of the quests in town with preset goals and objectives, and more of them out in the field and dynamic, more "discovered," so it feels more like an experience and less like a laundry list.
There's some of this in the game already. While killing Witherbark Trolls, you may stumble upon a Venom Bottle. While killing Firewing Elves, you may find a Druid in need of your assistance. But this type of questing in any large amounts really peters off after that initial Barrens and Ashenvale spurt, and the Alliance doesn't really get much of it at all.
The quest system itself is marvelous. It's a great way to focus on your character advancement and it's a far cry from early MMORPGs where there were almost no quests and all you had was the grind, but there's always room to grow. With WoTLK getting closer, some people have said that they feel that WoTLK needs something revolutionary to reel them in. Me, I say let's get a questing revolution in there, even if all that means is expanding on a great idea from a low level zone.