If you haven't reached 50 yet, don't plan on doing battlegrounds, aren't rich, and still want a different mount, you still can. Simply quest your way to exalted (and note that it's not exhalted or exaulted or some variant. People who spell it that way are rediculous and probably play rouges). This is also the best option if you plan on doing the Loremaster or Ambassador achievements. Lucky for you, it's not that hard, especially now that Blizzard has added the 'Low Level Quests' tracking type. All that's required is a knowledge of where to quest. Some factions are harder to get exalted with than others: Gnomeregan Exiles and Darkspear Trolls have quests primarily in their starting zones, then scattered all over Azeroth. Additional reputation for these factions will come primarily as 'spillover' from other factions' quests.
I've gotten exalted with Darnassus twice; once with a human and once with a Draenei. On both, I skipped a number of quests that gave primarily Darnassus reputation, and so didn't hit exalted until I was questing in Feralas. Of course, back then you didn't get your mount until level 40, so getting a cat at level 43 wasn't that long a wait. Now it's different, but there is hope.
Step 1 in getting exalted with a faction is identifying the faction you want to get exalted with. The next step is locating quest areas that give out plenty of reputation for your chosen faction. The racial starting area is a must. Quests there don't give spillover rep, but can be completed very quickly. There are usually a couple of other low-level zones that are primarily associated with your faction friends. Do every quest you can find in those zones.
I've mentioned spillover rep a couple of times now. Let me explain what I mean. Each quest you complete usually grants some experience and some reputation with a particular faction. Many (most?) quests for one of the primary factions also gives 'spillover' reputation to every other racial faction in your side, usually about one-fourth of the amount given to the main faction. For example, if you complete a quest that grants 250 reputation for Stormwind, you may also be credited with 62 or 63 reputation for each of Gnomeregan Exiles, Exodar, Darnassus and Ironforge. You won't see it in the chat panel, but you will see it if you keep track of your before/after reputation for each race. The primary exception to this rule is the racial starting areas. Quests in Elwynn Forest will usually only grant Stormwind reputation, quests in Teldrassil are Darnassus-rep only, etc.. The shared starting areas, Durotar and Dun Morogh, give reputation to both factions sharing the zone, but no other.
Let's say you want to get exalted with Darnassus. I know I do (again). You should first have completed your own starting zone. You might as well start out at level 10 or so. Once you're ready, head to Darnassus, run straight through to the other side, then follow the road all the way to Shadowglen. Enable 'low level quest' as your search type, run around and gather all the quests you can, then complete them. Turn them in and repeat until you're done. It might be worth consulting a quest guide to make sure you're not missing any. There are several guides at WoW-Pro or you can download their addon. You could also install Tourguide, Lightheaded and TomTom. These are all freely available tools that can help speed your questing without putting down a dime (other than your monthly fee). As you go on, you will find that there are a few out-of-the-way quests that you can easily miss without help.
Now that you're done Teldrassil, go do the quests in Azuremyst and Bloodmyst Isles. There are a fair number of quests that give Darnassus rep in both those areas, and many of the rest will give spillover. Don't miss the quests in Odesyus' Landing and Silvermyst Isle. They're a little out of the way, but give plenty of Darnassus rep. Especially that young girl that's hiding behind a log on the beach of Silvermyst. I've been there several times, but only found her once Tourguide pointed her location out to me.
Once you're done dealing with the Draenei, head over to Auberdine and do all the quests in Darkshore, Ashenvale, and Stonetalon Mountains. You will need a fishing skill of 30 to get the fishing quests from the gnome upstairs in the inn. Don't feel like doing them? Do them anyway. If you ever want to get exalted with Gnomeregan Exiles, you'll want the rep, and they definitely do grant spillover reputation. Also, once you've turned in the two 'Washed Ashore' quests to the NPC at the flightpath, be on the lookout for additional beached sea creatures all along the coast of Darkshore. Each one that you turn gets you more rep, and you'll be up and down the coast doing quests, anyway.
I've usually skipped Stonetalon Mountains and all the quests for Blackfathom Deeps. That was because I was in no rush. If you can wait until the 40s to get exalted, you can skip them too, and do Feralas instead later. I plan on doing them with my low-level shammy, though. I want to see how close to 30 I can get exalted.
Yes, it takes a lot of time to do these quests, but it's also going to take you a lot of time to do the battlegrounds or farm the cloth or gold for the other methods. This is also the only way to get another race's mount before level 50. Once you've done it this way, you'll also find that you've already made significant progress towards being Ambassador. If you want to explore all the beautiful areas of Azeroth, this may be for you. Unless you're Horde, in which case it's a great way to see all the brown and dirt that Azeroth has to offer.