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.
Each instance has a different number of enemies and it is the enemies that give the reputation, rather than the instance. The number of players in the party at the time does not influence the reputation gained so whether you solo the instance or not is irrelevant. In addition to this, the instances from The Burning Crusade expansion (Hellfire Ramparts, etc) will not benefit home faction standing at all.
Reputation gained is also affected by the following factors;
- If you are Human you will gain the 'Diplomacy' racial, adding 10% additional reputation
- Guild perks provide the 'Mr. Popularity' perk, which has two levels each adding 5% additional reputation (maximum of 10%)
In order to get solid numbers, the following questions need answers;
- What is your race
- What level is your guild
- Which instance are you clearing
Assuming a full clear, you should get the following reputation per clear of each instance, bare in mind that the instances with the highest reputation per clear may not be the fastest way to grind up a reputation, as this would need to take into consideration the time it takes to clear the instance:
Instance Name Reputation per clear
Ragefire Chasm 483
Deadmines 810
Shadowfang Keep Normal 548
Stormwind Stockades 515
Scarlet Monastery Graveyard 360
Scarlet Monastery Library 353
Scarlet Monastery Armory 287
Scarlet Monastery Cathedral 448
Uldaman 981
Scholomance 1261
Stratholme: Main Gate 737
Zul'Farrak 856
Stratholme: Service Entrance 917
Temple of Atal'Hakkar 2330
For the context of the question (fastest way to get reputation from Classic dungeons) I have excluded the long winded/windy instances (Wailing Caverns, Maraudon, etc) as what we're looking for is the fastest way to generate reputation. Seemingly the fastest instance to gain reputation out of the Classic instances is Sunken Temple. After the remodelling at the start of Cataclysm a large portion of this instance has been removed, leaving it as a single level instance capable of generating 2330 reputation per run which each run taking approximately 15 minutes on a level 60.
For completeness, here are the reputation values you can get per run out of a Wrath of the Lich King instance on heroic mode:
Instance Name Reputation per clear (HEROIC)
Ahn'kahet: The Old Kingdom 2300
Azjol-Nerub 1300
Culling of Stratholme 1700
Drak'Tharon Keep 2000
Gundrak 2250
The Nexus 2300
The Oculus 1600
Trial of the Champion 180
Violet Hold 1300
Halls of Lightening 2150
Halls of Stone 1850
Utgarde Keep 1950
Utgarde Pinnacle 2000
It is worth completely avoiding the likes of Trial of the Champion due to the lack of enemies present in the instance, clearing a combination of either Azjol-Nerub and Ahn'kahet: The Old Kingdom OR The Utgarde instances (Keep and Pinnacle) would probably be the fastest way to generate reputation, with these groups of instances having their entrances situated next to each other and being reasonably quick to clear.
Best Answer
If I remember correctly, tabards allow you to gain reputation from "any" instance (by "any" I mean probably only the ones relevant to that expansion set). I believe you can also get a small rep amount from doing PVP as well, but don't quote me on that, I could be thinking of PVP exclusive factions like Frost Wolves (horde).
Regarding your question: There are two instances that strike me as short enough to make "quick work" of the rep farming. The two are Lost city of Tol'vir, and Throne of the tides. The trash amount is quick to deal with, and some of the bosses can be skipped completely, like the crocodile boss in Tol'Vir. None of them are actually complicated and require minimum of brainpower, even if done in heroic mode. I've tanked, healed and been DPS in both respectively, both normal and heroic mode. Crowd control is useful for both places, a friend is good to have with you to support with that, unless you can plow your way through the content without a sweat.
Tol'Vir: It counts as "outside" so you can mount larger areas of the instance, making travel swift. Trash comes in packs of 3-4, rarely more. Can easily be CC'd and dealt with that way. At least one boss can be skipped. Relatively linear path to all bosses, pretty much a 'U'shaped path. Three of the four bosses summon adds to some degree, at least in Heroic.
Throne of the tides: Pretty straight forward shape of the instance, like a cross. One boss can be skipped, has an additional boss on heroic. Trash CAN be a bit brutal if you don't watch it, but those packs are few (big, tentacle dudes). The rest is simple. Bosses are complicated in the way that pure tank'n'spank isn't a really viable tactic.