So what are the easiest mounts to get hold of? First up, we have to look at the class-specific mounts, the warlock Felsteed and Dreadsteed, and the paladin Warhorse and Charger. Death Knights also get the Acherus Deathcharger, as well as being the only class to have the option to purchase the Winged Steed of the Ebon Blade. The paladin and warlock mounts are learned as abilities with the basic forms at level 20, and the latter ones at level 40. As DKs start later, they will get the Deathcharger through questing almost immediately, and can buy the Winged Steed at level 60.
Druids also have their travel form, and flight form, and Monks have the option to use the Glyph of Zen Flight, and of course, worgen get Running Wild but since these aren't really things you sit your character on, we're not sure if they count as mounts!
But other than those, what are the easiest mounts to get your hands on? Your home city, that is to say your faction's city, will have mounts for you to buy for gold. So undead can get mounts from Undercity, dwarves from Ironforge, and so on.
If you want to buy another city's mounts, though, you've got a bit more work to do. This is, however, an excellent way to get some more mounts fairly easily. You'll need to track down the city's quartermaster, for example, Magistrix Nizara in Silvermoon, and purchase a tabard. You can buy the tabard at zero reputation, and equip it, and this will earn you reputation whenever you go into a dungeon. It's a good move to go to another city and grab their tabard to start earning reputation as soon as you can, to make the best use of your dungeon time. These mounts are also account-wide, so make the most of them!
There are also guild mounts available, such as the Reins of the Kor'kron Annihilator or the Reins of the Golden King, which are the Horde and Alliance mounts rewarded for getting a guild to level 25, as well as for having exalted reputation with that same guild. You may want to obtain a guild tabard from the same vendor to boost your guild reputation. There are also mounts rewarded for guild raiding achievements, but they wouldn't be classed as easy to get if your guild don't have those achievements!
Lastly, there are several mounts available for honor. They're 2,000 honor per mount, requiring absolutely no reputation, or anything other than the points, so if you have the honor to spare, there's five good-looking mounts available.
These are mounts which, as the name suggests, drop from bosses, mobs or other things. Now there are some which are hard to get, either because of sensationally low drop rates, or because they drop from current raids. It's unlikely that anyone would recommend the Elegon mount as an easy one to get hold of.
Raid and Dungeon drops
However, there are many mounts that drop from older content, which is easily soloed, that could be considered easy to get hold of, if you are possessed of both patience and luck. The easiest one to get your paws on, by far, is the Bronze Drake. It drops from a special bonus boss, which is only available in the Heroic version of The Culling of Stratholme, if you complete the timed event in under 25 minutes. If you're at level 90, this should be pretty easy, as long as you know your way around the place. If you're fast enough, the Infinite Corruptor will appear, and the drake is a guaranteed drop.
Other than that, on the "easy" list there's also the Reins of the Raven Lord, which drops from Anzu in Sethekk Halls, the Swift White Hawkstrider which drops from Kael'thas in Magister's Terrace, the Ashes of Al'ar from Kael'thas, again, in Tempest Keep; the Deathchargers' Reins from Rivendare in Stratholme, and the Fiery Warhorse's Reins from Attumen in Karazhan. All of these are solo-able, but all require not only the level, gear and skill to solo the encounters, but also, thanks to low drop rates, a considerable amount of patience.
There are also mounts available from "Glory of the..." achievements, both in dungeons, which are easily soloed at 90, and old raids, which can usually be completed with a small group, so those can be relatively straightforward with some friends, but not alone! There is a complete list in the comments on Wowhead, so check it out!
There are also several rares that drop mounts. Zandalari Warbringers are the most recently added, which will need a small group to take down, and have a chance to drop a Direhorn mount, of a color that corresponds to the one they're riding. It can be very fast indeed to grind out all the mounts with a group of friends!
Other than those, there are countless mounts available from stalking, or, to use the proper term, camping a few rares. Whether these count as easy depends hugely on your patience, as a lot of these require a huge time commitment unless you're spectacularly lucky. There's a reason why it's called the Time Lost Proto Drake! There are many of these rares wandering the world, Aeonaxx in Deepholm drops a mount, Poseidus drops a seahorse.
There are so many of these mounts, as well as others that would fall into the above category, so as ever, Wowhead is your friend!
Again, I'm not sure whether these would be considered easy or not! The ease or difficulty of these is very dependent on the player's patience with grinding out reputation. For several reputations, it's possible to purchase a tabard and champion a faction in dungeons, and thereby gain reputation far faster.
This should be gone about in the same way as previously mentioned for city mounts, find the quartermaster for the faction, buy a tabard, and wear it. Ramkahen's camel mounts are very easy to get, for example, but any of them could be if you've intentionally or unwittingly built up reputation with a faction. A complete list of reputation-related mounts is available on Wowhead, and RepChk is a very handy tool indeed to display reputation items you have qualified for but don't yet own, just in case!
The mounts made with Tailoring have to be some of the easiest to get in the game, apart from the basic bought mounts. The basic Flying Carpet only requires a tailoring level of 300, and a few easily obtained materials. The downside with crafted mounts is that they often require any character riding them to be, for example, a tailor. The tailoring mounts all require this, which means both that they can't be sold on the Auction House, and that they aren't account-wide.
Engineers have it a little easier with their mounts, several of which can be sold in the auction house. Unfortunately, those are the ones with the more onerous materials requirements, so will likely end up costing a few thousand gold, at least. The easier engineering mounts are, unsurprisingly, only usable by engineers.
Mists added five mounts that can be created via Jewelcrafting, the four rares can be combined to create the Captain Planet-esque Jeweled Onyx Panther, but again, the materials requirement for these is pretty steep, so expect to either spend quite some time gathering them, or to spend a few thousand gold for one on the auction house.
Lastly, in this section there are two mounts discoverable by Archaeologists. The raptor mount is far more common than the Qiraji Battle Tank, and both require some considerable grinding to get the drop that leads you to them. What's more, Archaeologists can discover the recipe for the Vial of the Sands mount, but you'll need an Alchemist to make it, with, again, some fairly hefty materials requirements!
These are the last category we're going to look at today, and, again, a complete list of holiday mounts is available on Wowhead. Of the ones mentioned, the easiest to get are probably the recently added Darkmoon Faire mounts, simply because, rather than appearing only once a year, the Faire pops up once a month, allowing you to earn tickets at a more relaxed pace.
Of course, the ones that drop from a holiday-related instance, such as the Headless Horseman's Reins or the Big Love Rocket might be exceedingly easy to obtain, or never drop for you. A special mention goes out to the Magic Broom, which you will almost inevitably get during Hallow's End, but which can only be used when the festival is in town. Blizzard giveth, and Blizzard taketh away.
Good luck with your mount collecting, there's mount rewards in place for getting fifty mounts, which, if you're diligent with the above, shouldn't be too onerous a task!
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.