Each of the Taxi! quest chains requires that the character have access to the flight point of the connected major city. Characters of that race start with access to this flight point automatically, others must "discover" them. It is widely reported that these quests are restricted to the races whose starting areas are nearby.
You are asked to deliver an item (different for each quest chain) to the nearby major city. To get there, you are asked to talk to the nearby flight master.
Taxi!
You talk to the flight master. If you have not already "discovered" this flight point, you do by talking to him now. He instructs you to purchase a flight from there to the major city.
You do this. When you arrive, you seek out the person you were instructed to contact.
Picking up the package
Your contact hands you an item to be returned to the person who sent you off on this mission, by way of the major city's flight master.
Return delivery
The city's flight master, like that of the rural one you'd talked to before, instructs you to buy a flight (back to where you came from).
You do this, and are rewarded for your efforts.
Rewards
You are given close to 5 in quest rewards, along with a varying amount of reputation. You earn a good bit more than the use of the taxis cost.
Notes
The Night Elf Taxi quest is structured slightly differently, in that the first quest directs you, essentially, "how to get out of Darnassus to the rest of the world". The last two quests are ended and begun (respectively) by the second NPC, instead of the flight master. As well, that chain is unique in that you start in the major city, instead of the smaller town.
The quest chain Ironband's Blastpowder, similar to the Dwarf/Gnome Taxi! chain, has you going from Loch Modan to Ironforge and back, but on a more extended errand.