Levels 1-6 (Valley of Trials)
01) I do every single quest in Durotar! Here's the fastest way to do em:
02) Start off accepting "Your Place In The World". (Right in front of you)
03) Turn it in (at 42.68) ... accept "Cutting Teeth"
04) Go start doing: "Cutting Teeth" (keep step #5 in mind)
05) Once you hit level 2, go accept "Sarkoth" (at 40.62)
06) Go do "Sarkoth" (at 40.66).
07) Turn "Sarkoth" in (at 40.66) ... Accept "Sarkoth" pt.2
08) Make sure "Cutting Teeth" is complete.
09) Go turn in "Sarkoth" pt.2 and "Cutting Teeth" (at 42.68) ... Accept "Etched Tablet" and "Sting of the Scorpid"
10) Turn in "Etched Tablet" (around 43.69) and get new spells/abilities for your class.
11) Accept and do the following: "Sting of the Scorpid" "Vile Familiars" "Galgar's Cactus Apple Surprise" and "Lazy Peons" (all these are done north and north-east of Valley of Trials).
12) Turn all those quests in, then accept "Burning Blade Medallion" and "Thazz'ril's Pick"
13) Go do: "Burning Blade Medallion" and "Thazz'ril's Pick" (these are done in the cave at 44.56)
14) Once those two quests are done use your hearthstone.
15) Turn those 2 quests in and accept "Report to Sen'jin Village"
16) The starting area is now completed.  Leave starting noob zone (by heading east).

Levels 6-9
01) Accept "A Peon's Burden" (at 52.68)
02) Accept "Thwarting Kolkar Aggression" (at 54.75)
03) Go turn in "Report to Sen'jin Village" (at Sen'jen Village, 55.74)
04) Collect all the quests in Sen'jin Village. (which include: "A solvent Spirit" "Practical Prey" "Minshina's Skull" "Report to Orgnil" and "Zalazane")
05) Do NOT do "A solvent Spirit" at the water around Sen'jin Village. Make sure the quest is accepted though.
06) Run up to Razor Hill (52.44), grinding mobs along the way, and make sure you get to level 6 before you get up there for new spells.
07) Turn in "Report to Orgnil" and collect all the quests at Razor Hill: "Dark Storms" "Vanquish the Betrayers", "Encroachment" "Break a Few Eggs"
08) Go up to the watch tower north-west of Razor Hill, at 49.40 and accept "Carry Your Weight"
09) Go in the Inn (51.41) and turn in "A Peon's Burden".  And also make Razor Hill your home.
10) Get first aid. (at 54.41)
11) Go do: "Vanquish the Betrayers" along with "Carry Your Weight" (these are done at Tiragarde Keep, 57.55)
12) After killing Benedict (he is up in the building at 59.58) and get his key, go up the steps, then open the chest and grab the note that starts: "The Admiral's Orders" accept the quest.
13) Go turn in "Vanquish the Betrayers" "The Admiral's Orders" and "Carry Your Weight" at Razor Hill (52.44), and accept "From The Wreckage...." "The Admiral's Orders" (part2).
14) Go do "A solvent Spirit" and "From The Wreckage....".  Do these two quests at the water east of Tiragarde Keep (around 62.50).
15) Do the first half of "Encroachment" (at 49.49).
16) Then if you're not at least 3 bars away from lvl 8, grind till you are, go turn in "From The Wreckage...." at Razor Hill (52.44) and get new spells/abilities.
17) Run way down south and do "Thwarting Kolkar Aggression" (at 48.79)
18) Once that is done turn it in (at 54.75).
19) Turn in "A solvent Spirit" at Sen'jin (55.74)
20) Go to Echo Isles (the islands south-east of Durotar) and do the following 4 quests together:
21) "Break a Few Eggs"
22) "Practical Prey"
23) "Minshina's Skull" (the skull is at 67.87)
24) "Zalazane" (Zalazane is at 67.86) (See Video)
25) After completing all these quests, die on purpose, so you end up right at Sen'jin Village (55.74)...
26) Turn in "Practical Prey" "Minshina's Skull" and "Zalazane".  Save the quest reward item Faintly Glowing Skull for a later quest called "Burning Shadows"
27) Hearth to Razor Hill. Turn in "Break a Few Eggs"
28) Do the second half of "Encroachment" (west of Razor Hill at 42.38)

Levels 9-12
01) Go up and accept "Lost But Not Forgotten" at the little hut at 43.30
02) Go up (grind mobs along the way) and accept  "Winds in the Desert" at the goblin, 46.22.
03) Do "Winds in the Desert"
04) Turn it in, accept "Securing the Lines" (at 46.22)
05) Grind mobs to lvl 10.
06) Go down to razor hill (52.44), turn in "Encroachment"
07) Get new spells/abilities.
08) Do your level 10 class quests. I do this as a hunter: do all three pet quests "Taming the Beast":
09) Dire molted bore (south of razor hill)
10) Surf crawler (north east of razor hill at 58.30), DON'T go down to Senjen Village, it's too far away. Then Hearth if you can, if not run back.
11) Armored scorpid (north west of razor hill)
12) After completing all those, accept "Training the Beast".
13) Go way up north-east, (at 57.16) and tame an Encrusted Surf Crawler for your first real pet.
14) Go way west (grinding mobs along the way) and accept "Need for a Cure" (at 41.18)
15) Go to Orgrimmar (at 45.11)
16) Turn in "Training the Beast" at hunter trainer. (66.18 in Orgrimmar)
17) Turn in "The Admiral's Orders" in thrall's chamber (34.36)
18) Accept "Hidden Enemies" (Thrall gives it)
19) Go into Cleft of Shadow, accept "Finding the Antidote" (46.53)
20) Go do: "Securing the Lines" (at 53.23 in Durotar)
21) Turn in "Securing the Lines" (at 46.22)
22) Go do: "Finding the Antidote" (the scorpids are all over the place) and "Lost But Not Forgotten" (all along the river west of Durotar)
23) Go do: "Dark Storms" (at 41.26)
24) Turn in "Lost But Not Forgotten" (43.30)
25) Turn in "Dark Storms" (at Razor Hill, 52.44)
26) Accept "Margoz"
27) Turn in "Margoz" (at 56.20) accept "Skull Rock"
28) Go do "Skull Rock" along with "Hidden Enemies" (in the cave at 54.11)
29) If you can, try to kill Gazz'uz, he drops 'Eye of Burning Shadow' item which starts "Burning Shadows" quest. TIP: use the quest reward item Faintly Glowing Skull to help you kill him, he's tough without it, also try to party up with somebody if you can. (See Video on how I killed Gazz'uz)
30) Turn in "Skull Rock" (56.20) accept "Neeru Fireblade"
31) Go into Orgrimmar.
32) Turn in "Hidden Enemies" (33.37 in Orgrimmar) accept the next part to "Hidden Enemies"
33) Go to Cleft of Shadows (47.53) and turn in "Finding the Antidote"
34) Turn in "Neeru Fireblade" and "Burning Shadows" at Neeru Fireblade (49.50) accept "Ak'Zeloth"
35) Also keep talking to him to complete "Hidden Enemies"
36) Go back to Thrall and turn in "Hidden Enemies" again. The 3rd part refers to RFC instance, which I skip.
37) Leave Orgrimmar, and turn in "Need for a Cure" (at 41.18) (you will most likely have to accept it again to turn it in)
38) Hearth to Razor Hill.
39) Build up first aid.
40) Get new spells/abilities.
41) Accept "Conscript of the Horde" (at 50.43 in Razor Hill)
42) Run west into the Barrens...