If you don't want to read this entire article, but you're primarily focused on how to get your mount at level 20, follow these steps. They will work for anyone, require no grinding sessions, and generally offer a high reward-to-time ratio:
If you do nothing else, do this: Learn to use the Auction House, both for buying and selling.
Consider installing the Auctioneer addon and use it to your advantage in the Auction House.
Bargain hunt, do not pay full price.
Always post a buyout price on your auctions - many players will not bid on an auction with no buyout, unless the item is heavily discounted, and then it usually sells for a fraction of what it could have gotten. You will have more sales at a higher price and get your money more quickly if you post a proper buyout price.
When you have saved around 5 (and you will quickly, due to having taken gathering professions), you can start investing. Buy cheap items on the AH and then re-list them for a profit. Only buy and sell what you know, and test your theories in small quantities.
Take one or (better yet) two of the gathering professions: Mining, Herbalism, or Skinning. All of them are surefire moneymakers. Mining is said to bring in more money at the cost of being slightly more difficult to farm, while skinning is simple but time-consuming and can fill your bags very quickly.
While you don't have to Fish (and fishing is not to everyone's taste, as it involves a lot of sitting and staring at a bobber), fishing can become a good source of income as you level.
Wait until you are level 40, or even level 70 to take crafting professions—they can be money sinks and you can buy the items you would have purchased with the extra money from gathering. Raw materials are usually worth more than a crafted product.
If you do craft, learn what sells - large bags are astonishingly valuble when an update is due to come out.
Generally, do not craft white items unless you know there is a demand for that item.
Learn what stats are useful and craft items that appropriately enhance those stats.
Do not overproduce; increased supply depresses price. You will get a better price per item if you sell fewer of an item.
Be frugal upgrading your gear.
Every bright, shiny new piece of gear is going to be old and shabby in a level or two. If you socket gems, you'll lose them when you upgrade your weapon.
Learn what stats help your character, and stick to that gear - avoid having to replace gear.
You do not need to upgrade at every opportunity, especially while leveling.
Does the gear you are thinking of upgrading to offer a significant improvement? Think in percentage increases.
Plan on getting gear from quest rewards, drops, or instances.
Remember that as you level, you will be replacing your gear soon, no matter how good it looks now.
When leveling, avoid investing a lot in high-priced gear, especially blue or purple gear. You do not need rare or epic gear to level.
Remember that the demand for blue and purple gear is driven primarily by twinks and alts. These are players that have substantial financial resources already, and can afford to pay a premium. As a result, you will not find many bargains in such gear; they tend to be vastly overpriced.
Participating in raid groups into instances can get you the same gear.
If you're a first-time player, learn first. Research. Ask. Test. Try, wisely. And above all, always spend cautiously--you'll need that money later.
If you follow this basic advice you should have no problem at paying for the mount at level 20, and you will always be able to afford skill training, food and potions along the way.