As a Priest, you'll want to mainly cast Flash Heal, keep Renew active on the tank and only shield when necessary before reaching leveling 60. 60 is where you get your power heals so you may learn what rotation you feel most comfortable with but try not to use AoE heals too often, in addition to the occasional Holy Nova or Circle of Healing when you feel it is safe enough. At 60 you get Penance and will love your heals after you receive this spell, it really is a beauty.
As a Druid, you'll want to mainly cast Regrowth on a target with 75% HP, instantly rejuvenate them and should barely use Healing Touch as it isn't mana efficient and takes far too long to cast before dinging level 64. After level 64, you'll be doing pretty much the same thing but every now again throw Wild Growth up, and keep Lifebloom on the tank when he's fighting bosses. Nourish is learned at 80, and it will soon become your best friend, it's the equivalent wow powerleveling to a Priest's Flash Heal.
Let's begin by saying that you should always make sure that you have collected all the quests associated with the instances you are planning to run, as they grant a considerable boost your overall leveling experience. It must be said that choosing a faction best suited to a healer is now impossible to do since both now have Paladins and Shamans. Before the BC, it may have been a debate but now it's null and void.
For Alliance, you can solo quest or find a partner to get yourself to level 16, from there mass instances in between zones. Start healing the Deadmines at 16; quest around Westfall and the Wetlands and instance until around level 24, than heal Shadowfang Keep, Blackfathom Depths, and the Stormwind Stockades. This should get you to about level 27. Quest in Duskwood and heal Gnomeregan until you're about level 30. Once you ding 30 you should go to Desolace and try to get a group for the Scarlet Monastery, either the Library or Graveyard wing. This should get you to around level 34, once you ding, you can head to Stranglethorn Vale while occasionally running the Scarlet Monastery's Armory and Cathedral wings. You may want to consider running Razorfen Downs, allowing you to reach level 39 or 40.