Be aware of all quests for an area before going there, which will involve some out of game research. It is more efficient to spend a little time out of game than to blunder around in game. You will also get more reputation from doing quests before they turn green.
If you can, aim to grind on yellow mobs, ones that are your level or one or two levels below you. This seems to be what most classes are designed for. A good grind spot has a lot of relatively static spawns; ones that roam over a wide area are likely to generate adds. Only group for instances and elite quests – which you do only to relieve the boredom.
Start in the newbie area (usually a subsection of the starting zone), and do all quests. Mobs are non-aggressive. Be aware of what the starting quests (and class specific quests) are, and who gives them, so that you can stack them efficiently. Stay to at least level 5, possibly 6; grind if you have completed the quests.
Next, rotate through all of the reachable starting zones for your race. Most of these will have a levelling gap in them, where the mobs for the next quest are marginally tougher. When you hit this, move on to the next zone. The starting quests in most areas will usually be a couple of ‘meet and greet’ quests (aka ‘runarounds’), with possibly one or two ‘kill x’ quests. Alliance characters have a disadvantage compared to Horde, as the starting area for night elfs is hard to reach; the tauren area is much more accessible for a low level Horde character.
By the time you have completed all of the accessible starting areas, you will be 12. Repeat the same cycle in secondary areas, again until you reach a levelling gap.