Below are listed a few Alliance-favoring areas and quests you might want to visit and do while leveling. Why? Simply because in these areas and for these quests either the rewards are really big, the quests are really easy, the quests are really close together, or you can get several quests done by killing the same group of mobs. Certain areas are also really good for a specific professions, or rare drops.
This article is intended as a leveling guide. This article is not intended as a Places to Farm list to get certain materials however when certain creatures listed in these quests give notable items, for example the elemental earth in badlands, it is helpful for some players to have it included. This article is not intended as a list of places to grind. Basically, If you ever thought something like "wow, i wish i knew beforehand that i could do 'this,this and that' <here> together" you probably found a notable area and you're encouraged to expand the list.
1 to 10 Edit
The beginning areas are rather simple and well-defined. The Night Elves have Teldrassil (the big tree), the Humans have Elwynn Forest, the Draenai have Azuremyst Isle and the Dwarves/Gnomes have Dun Morogh. The questlines are very straightforward, beginning with the person right in front of the newly created character and continuing with a few added quest lines from there. Levels 1-5 can be finished in under an hour even by the most inexperienced player and should involve few, if any, deaths. It is highly advisable to do these quests as well since they give basic money and the opportunities to both a) learn the game and b) the chance to gain slightly better armor than the starting gear. One note, however, while the starting gear is enough to level to about level 5 it is quickly replaced. Most toons are very poorly equipped to begin with (although the creatures they are fighting are equally as weak to reflect this) and the gear that they pick up along the way - even such basic gear as grey "vendor trash" - is an upgrade (albeit not fantastic).
For starting players: grey gear is called vendor trash because it is generally sold to a vendor right after it outlives its usefulness (or in some cases is never used because it has inferior statistics, but is worth cash to vendors and can help the players pay for repairs and other things). White gear is better than grey gear because it (usually) has better statistics. Green gear is known as uncommon gear and some green gear is fabulous, while other green gear is not very good. Typically it depends on the toon's level, with each toon getting better greens (and blues etc. below) as they gain levels. Blue gear is rare gear and generally has either a) some kind of rare quality that makes it exceptionally good (such as very high statistics, a lot of damage or healing or occasionally the ability to do something special) or b) it sells for a lot of gold. The first pieces of blue gear are available in dungeons known as "instances" and are available for use around level 12-13 or so. Purple gear is called "epic" gear and is generally only available to really high level toons. It is considered the highest quality gear and often is the best in the game, but requires quite a bit of time to be able to get. Also, occasionally a toon has "gold" gear which is a Bind to Account item, which means that they have a high level toon (usually maxed out) feeding them. This gold gear is the equivalent of a blue item and it levels as the toon does.
10 to 20 Edit
There are a couple of areas that are for levels 10-20. The Draenei can continue in Azuremyst Isle/Bloodmyst Isle (they are connected) up to level 24 or so, or they can go to Darkshore (a zone they share with the Night Elves). There is also a boat for the Alliance to take to Stormwind if they should choose to go to the human zone of Westfall (where some very powerful Alliance only items can be obtained through a series of quests). The dwarves can continue to Loch Modan, or they can catch the tram connecting Stormwind to Ironforge and go to Westfall as well.
Also, The Redridge Mountains are connected to Elwynn Forest on their eastern side. This zone is for players level 15-25 and can supplement those toons who are questing in Westfall.
Notable Westfall quest items: The Tunic of Westfall, The Staff of Westfall or The Chausses of Westfall