Well for your first character quests are no doubt the fastest way of leveling. Go into a zone for your level (you can find zone levels on the WoW site) and go to the main area of your faction in that zone. Pick up all the quests you can find there. Go and finish all of them and come back. A ton more will now be available for you, and do all those. Keep doing this pattern and moving from zone to zone. If you have spare cash you can also pay for dungeon runs from an 80, but this can get a little pricy for a first character and you miss the whole questing aspect of the game. Grinding is said to be good, but I have tried it and it it much slower than questing.
Also always stay in LFG for 3 good dungeons for your level while you are questing. If you happen to find a group, join it but keep questing until you have a full group, in case the group breaks up. Dungeons are a good way of leveling, making money (although quests get you money much faster), and gearing up.