This is the first of a series of leveling guides for druids. Because you learn your animal forms at such widely spread levels, I will be writing a guide for each section of your life. You learn bear form at lv 10, Aquatic form at lv 16, cat form at lv 20, cheetah form at lv 30, dire bear form at lv 40, and bird form at lv 68. You can also learn tree form or moonkin form when your talent trees allow.
This guide covers just the first stage of life, when all you have is caster form. From level 1-10 you are limited mostly to your damage and healing spells. You can still kill reasonably effectively in this form. Start as far away from an enemy as possible, then fire off a wrath spell at it. The second it flies off, fire another. If you time it right you can get a good chunk of its health away before it even gets to you. Then moonfire it and start hacking away at it with your melee weapons. Pop on a rejuvination spell when you're hurt and keep going. When you're out of mana, drink up and go back to blasting. Once you have entangling roots, you can root enemies when they get too close, run a good distance away, then fire away again. This method of killing is probably the slowest because you constantly have to replenish your mana. Thankfully you recover very quickly up to about level 10.
Don't worry too much about gear at this point as you will be outgrowing it so fast. If you want to boost your stats, intellect and spirit would probably be the best to go for. They will increase your max mana and mana regeneration rate so you spend less time drinking and more time killing. Once you hit level 10, go to your trainer at Darnassus and start up on the quest to earn bear form.