Druids rule in the World of Warcraft and that's basically it in a nutshell. Start a Druid and you can be Tank, Healer, Ranged, or Melee Damage (DPS) with one character and you'll fill all those roles at least as well as any single class character. Heck, you'll be better because you can switch back and forth as necessary. That Rogue may by a bit ahead of you in damage done, but she can't be an emergency tank or healer and her bolt-lobbing capabilities are nil.
In PvP, well played Druids can be very hard to kill. They hit as hard as anyone and ambush almost as well as Rogues. Combine that with heals and the ability to escape all entaglements, by shapeshifting, and they become formidable PvP machines.
These days, in the Mists of Pandaria expansion, leveling a Druid is as easy as it's every been. The in-game quest helper shows you where to go and it helps with the "what to do part." It's not a real leveling guide, but it's a lot better than what you had to do when WoW began. Add in PvP and the dungeons and you can be level 90 in minutes. Well, er, um... a few days. Either way it's faster than ever before.
Leveling Tips
Now that that's out of the way, here's some info on leveling your Druid as quickly as possible. Stick with the Feral Cat spec for now as It will serve you very well while questing, your damage in the dungeons will be just fine, and you'll eat enough people in PvP to stay happy. (Guardians are fine, but damage is low, and Balance just isn't as effective for leveling.)
As a Feral Druid you will gain a set of abilities unique to your specialization (spec.) Mists of Pandaria changed things greatly from previous expansions and it's no longer the talents which set you apart from the other Druids. Any Druid can take any of the talents, but certain talents work better for you than others.
So we're looking for utility here, but these aren't the only choices. One of the interesting things about the new talent system is that you're not locked into your picks. You can change them at any time and all it costs you is a cheap reagent from the Reagents vendor or Inscription Supplies vendor. Just pick the new talent you want, from any one row, pay the reagent cost, click the "learn" button, and you have the new pick.
Gearing up
What about gear and stats? Gear's pretty easy. For weapons you want the highest DPS number possible, then go for Agility and Stamina. If you stack any stat go for agility, especially as you get higher level. It's easily your best stat. Don't worry about the other primary stats (Int, etc) as you level, they're not very useful. Hit and crit have value, though. I recommend skipping enchants and gems as you'll level out of them quickly and they're expensive for the value they provide.
Potions of healing are very nice to have while leveling your Druid. Stick them on you action bar. You'll get an instant heal without having to leave cat form. You can also obtain potions, scrolls, and foods that will buff stats. These are all useful and will help you level that much faster, either by increasing damage or lessening downtime.
Always carry the biggest bags you can afford, so that you don't often have to go back to town to sell stuff. Bags can fill up quickly when doing a lot of hacking and slaying. Speaking of towns, always log out in an Inn to get the benefit of the "rest XP" when you return.
Try and group your quests. You will have several in one area while others will be "over there" somewhere. Do all the closely grouped quests before moving away. Always think about your path and plan out the most efficient route, you can save a whole lot of time that way.
Ok, I'm starting the timer now. Let's see how fast you can get your Druid to 90.