Past level 80, the quickest way to is take the portal from Stormwind to Uldum and simply fly to Tanaris. Otherwise, take the Tram to Ironforge then fly (or walk) to Menethil harbor. From there, take the boat to Theramore and swim through Thousand Needles (south of Dustwallow Marsh) and right into Tanaris.
Or take the ship from Stormwind to Northrend, fly to Dalaran, take the portal to the Caverns of Time.
You can also fly from Stormwind to Stranglethorn Vale and go to Grom'gol Base Camp. This is a horde camp so be careful.
Stand on top of the zeppelin to Orgrimmar. Ride it and then fly from Orgrimmar to Tanaris which is a bit faster.
Note this is a dangerous way of travelling to many places of Kalimdor, but is arguably the fastest one.