If this is your first time playing World of Warcraft, stick with PvE. Player vs Environment will give you plenty of interest to deal with and you won't have to worry about other players coming up and ganking you from behind while you are just out and about, minding your own business. Once you have more experience and are playing an alt character, have fun with PvP. You'll probably level even faster.
1. Complete all your quests.
In your Druid starting area do not leave until you have completed every single quest! Levels 1-10 are the absolute easiest and fastest levels in all of World of Warcraft and these early levels equip you with lots of useful stuff. If you skip a quest here and miss out on something crucial you'll need later, you'll regret it. So do all the quests, get as many spells as possible, get as much money as you can (believe me, you will need it!) and make sure you get your Bear shape shifting skill at level 10.
2. Start training for your professions early.
Early training will help you keep your profession's level on par with your overall level. Good Druid professions are herbalism, skinning, alchemy and leather-working. Leather-working will cost you more money to level it, so you may want to skip it.
Combining herbalism with skinning, on the other hand, should make for a nice profitable combo as you can sell raw materials and more wow gold is always better. If you can buy the items you need for training easily, you will naturally level faster as a result.
3. Use the Feral spec tree for your Druid.
The feral specs make the most sense for solo leveling and grinding and its your best choice for PvE. While the Druid is a hybrid, don't try to over-balance. You'll just wind up having to carry a lot of gear around, a lot of gear that you had to spend money on. Get some healing talent in there, but otherwise focus on Feral.
4. Consider an alternative starting area.
Druids normally start off in the Night Elf starting area for Alliance and the Tauren starting area for Horde, but just because your race is predefined once you pick Druid, that doesn't mean your starting zone has to be. You can choose another zone.
Hopefully, these tips will help you move through levels 1-20 in particular very fast, but there's more you can do to ensuring smooth sailing from level to level. You'll find that a wow power leveling guide will really help as they provide step-by-step walk-throughs for the most efficient game play.
