If we asume you got no other transportation device then your flying mount, you'd simply jump on the "shark" Zeppelin to Borean Thundra and fly from there.
If you visit Dalaran rather often, I strongly suggest you to place your Hearthstone there, since there's a portal to Orgrimmar in Dalaran.
Other ways of getting to Dalaran; the engineering Worm Hole which allows you to chose one of a couple of randomly given zones of Northrend, which is rather close to Dalaran itself. Avoid choosing Howling Fjord and Borean Thundra since they're the most distant ones from Dalaran.
If you've grinded like a madmen with Argent Tournaments in Ice Crown, you can use their tabard to teleport there.
There's no point mentioning warlock summoning, but there's also the guild ability Have Group, Will Travel which allows someone positioned in Dalaran to summon you.
If you got silly much money and for some reason don't like to pay mages a small sum, you can on your next visit to Dalaran purchase an expensive ring which allows you to teleport to Dalaran.
I asume that you're a high level since you ask for the fastest way, and not the safiest way.
To start, as a level 1 you should be able to make it to and from all of the major racial Azeroth cities with little issue. Generally you will find a boat or other means of "long" transport (e.g. the tram from Stormwind to Ironforge).
Mage Portals & Warlock Summons
Additionally there are no level requirements attached to the Mage's portals or the Warlock's summon ability. This means, assuming you have a Warlock that is at least level 42 (when the spell is available) and 2 willing friends near him to help him summon you, you can literally go anywhere that the Warlock is willing to go first. If you have a Mage friend, you can go anywhere he can make a portal for. You can find the portal locations available to a mage on the Mage Portals wiki. As you might assume these portals include your standard capital cities of that Mage's faction as well as places like Shattrath in Outlands, Dalaran in Northrend, Vale of Eternal Blossoms in Pandaria and to answer your specific question, his faction's city in Ashran (Stormshield and Warspear respectively).
Finding a Mage willing to make a portal for you is your easiest means of travel as it only requires one person opposed to a Warlock basically needing himself and 2 additional members to both summon his ritual stone and then to help summon with said stone. With enough money you should be able to talk a Mage into doing this for you in trade chat, which is opened anytime you are in a major city.
Capital to Capital Travel
They've made the capital cities for each faction incredibly connected, whether you take a boat from one to another or often they simply put portals. They've also set up portals to the expansion zones directly from the cities, however these portals usually don't appear until you've either completed a certain quest (of the appropriate level) or they could potentially be level locked.
Opposing Faction
As for the idea of getting to an opposing faction's capital city, this is potentially more difficult and will require some foot-travel in most cases as you would have to make your way to the closest place that your faction controls near your destination then go by foot or mount from there to your desired location. I'm not sure what you'd hope to accomplish if this was your mission as a level 1 however, it would likely be chain deaths and would take hours. Naturally any portals to the opposing faction's city will either simply not appear to you or will be faction locked.
I didn't go into any real specifics, I'm sure you can find the right boat for the job if you're plan is to start a character in one place and then travel to another to level. If I knew more about what you were trying to accomplish I might be able to get you more specified information.
Best Answer
Assuming you aren't a mage and can't teleport yourself:
The fastest way to Dalaran is to teleport yourself with one of the sixteen rings of the Kirin Tor. If you played during Wrath, you might still have one laying around in Void Storage. If not, well, they start at 6800 gold (at Exalted with the Kirin Tor). Visit Harold Winston to buy one.
You can also use Jaina's Locket to create a portal to Dalaran on a one hour cooldown. This can sometimes be found for 10K gold or less on the auction house, or you can run the Shadowmourne legendary quest chain and get your own.
The second fastest way is to ask in chat while in Ashran or your garrison and buy a portal to Dalaran from a mage.
If you played in the Argent Tournament, you can use their Argent Crusader's Tabard to teleport to the tournament grounds, and Dalaran is a short flight away.
If you're an engineer, you can use a Wormhole Generator: Northrend to get to Northrend, and then fly from Storm Peaks to Dalaran.
If you're a monk and can fly in Pandaria, you can take a Zen Pilgrimage to the Peak of Serenity, then fly to the Shrine, then take a portal to Dalaran.
Failing the above, my suggestion would be to take the portal to Stormwind/Orgrimmar, then the portal to Pandaria, a flight to the Shrine, and then the portal to Dalaran within. This should be faster than taking a ship to Northrend as you won't have to wait for it, and the flight in Pandaria is shorter than the flight in Northrend.