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From Stormwind, go to Stormwind Harbor, then take a boat (on left) to Darnassus. Find flight master and fly to Darkshore. From Theramore Isle, swim north along the coast, then go west (safer than going through Dustwallow Marsh). From Ashenvale, go east of Astranaar and then south.
First of all it should be mentioned that if you want to reach some point on the map while you're still low level, and the way is too dangerous, your best bet is to have a mage friend who can portal you to the capital cities for your faction. That should get you closer to where you want to go.
Any Alliance mage level 40 or higher should have the portal spells for Ironforge and Stormwind. However the portal to Darnassus is only available to mages level 50 or higher.
If you want or need a travel route added to this guide, put your request in the discussion area.
General
The Alliance homelands are divided between Teldrassil (island northwest of Kalimdor) and the southern subcontinent of the Eastern Kingdoms.
On Teldrassil, you'll find Darnassus, the Night Elf capital on the western part of the island. The night elves also maintain major outposts in Darkshore (Lor'danel) and Ashenvale Forest (Astranaar)
On the eastern continent, the Dwarf capital of Ironforge is located at the far northernwestern part of the southern sub-continent in Dun Morogh. In addition, the dwarves maintain outposts in Loch Modan (Thelsamar), and Wetlands (Menethil).
Also on the eastern continent, the Human capital of Stormwind City is located at the western part of the southern sub-continent in Elwynn Forest. In addition, the humans maintain outposts in Westfall (Sentinel Hill), the Redridge Mountains (Lakeshire), and Duskwood (Darkshire).
How To Get To Each Region
Eastern Kingdoms
Elwynn Forest / Stormwind City
From Dun Morogh, go to the Tinker Town in Ironforge, then take the Deeprun Tram to Stormwind City.
Dun Morogh / Ironforge
From Elwynn Forest, go to the Dwarven District in Stormwind City, then take the Deeprun Tram to Ironforge.
Tirisfal Glades / Undercity
From Silverpine Forest, go north or swim across Lordamere Lake from Dalaran.
Loch Modan
From Dun Morogh, go east.
From Menethil Harbor in the Wetlands, go east and then south.
Silverpine Forest
From Hillsbrad Foothills, go northwest.
Westfall
From Elwynn Forest, go west.
Redridge Mountains
From Elwynn Forest, go east.
Duskwood
From Elwynn Forest, either swim across the southern river, or go east until you reach the crossroads leading to Redridge Mountains and Duskwood.
Hillsbrad Foothills
From Arathi Highlands, go northwest through a passage in the wall.
Wetlands
From Dun Morogh, go east to Loch Modan, then north.
From Teldrassil, take the hippogryph route from Rut'theran Village to Auberdine, then take a boat to Stormwind Harbor, from there the Deeprun Tram to Ironforge, then walk through Dun Morogh to Loch Modan and from there to Wetlands
Alterac Mountains
From Hillsbrad Foothills, go north.
Arathi Highlands
From Wetlands, go north through Dun Modr and Thandol Span.
Stranglethorn Vale
From Darkshire in Duskwood, go southwest.
Badlands
From Wetlands or Dun Morogh to Loch Modan, then go south.
Swamp of Sorrows
From Elwynn Forest to Duskwood to Deadwind Pass, then go east.
The Hinterlands
From Arathi Highlands to Hillsbrad Foothills, then go northeast (behind Durnholde Keep).
Searing Gorge
From the Badlands, go west.
The Blasted Lands
From the Swamp of Sorrows, go south.
Burning Steppes
From the Searing Gorge, go south through Blackrock Mountain, around the perimeter of the molten span, to the south exit.
Western Plaguelands
From Hillsbrad Foothills, swim the river to the northeast.
Eastern Plaguelands
From the Western Plaguelands, go east.
Kalimdor
Durotar / Orgrimmar
From Theramore Isle, swim north along the coast.
From Ashenvale, go east of Astranaar and then south to the Barrens, follow the Gold Road south, then cross the east river separating the Barrens and Durotar.
From Azshara, swim Windfury River south, then follow ashore east.
Mulgore / Thunder Bluff
From the southern Barrens (accessible from Dustwallow Marsh), go west.
Teldrassil / Darnassus
From Auberdine in Darkshore (see below), take a free hippogryph ride to Rut'theran Village.
NOTE: If you are not a Night Elf and traveling to Teldrassil / Darnassus for the first time, you will not be able to use the hippogryph. You can take a boat from the northern dock of Auberdine to Rut'theran Village.
Darkshore
From Stormwind, go to Stormwind Harbor, then take a boat (on left) to Darnassus. Find flight master and fly to Darkshore.
Barrens
From Theramore Isle, swim north along the coast, then go west (safer than going through Dustwallow Marsh).
From Ashenvale, go east of Astranaar and then south.
From Azshara, swim Windfury River south, then follow ashore west.
Stonetalon Mountains
From Ashenvale, take Talondeep Path in the south, east of Mystral Lake.
Ashenvale
From Darkshore, go south.
Thousand Needles
From the Barrens, go south.
Desolace
From the Charred Vale in Stonetalon Mountains, go south.
Dustwallow Marsh
From Dun Morogh, reach Menethil Harbor in the Wetlands, then take a boat to Theramore Isle.
Feralas
From Desolace, go south.
Tanaris / Gadgetzan
From Theramore Isle, swim south, drown when you get into Tanaris waters, drown, resurrect there.
Alternatively, make your way through Feralas to New Thalanaar, do 1 mission to get a boat to the SpeedBarge in Thousand Needles, then from there swim straight south to the shore and head south to Tanaris.
Azshara
From Ashenvale, take the road east.
Felwood
From Raynewood Retreat in Ashenvale, take the road north.
Un'Goro Crater
From Gadgetzan in Tanaris, head southwest through the desert. The Crater is west of Tanaris. DO NOT jump.
Winterspring / Everlook
In Felwood, become not hostile to the Timbermaw Furbolgs. Then travel through Timbermaw Hold in the north of Felwood (alternatively, can be accessed from Moonglade). Take the East tunnel to Winterpring.
Moonglade
Non-druids: from northern Darkshore, swim east until you get into Moonglade waters, drown, resurrect there (safest).