OK, questions has been asked many times.
Here is what you do while leveling:
Do Battlegrounds, as A) It gets you honour but B) It now gets you experience points for completing objectives which means your "killing 2 birds with one stone" as the expression goes.
DO NOT spend your Honour Points whatever you do. Wait until you get to level 80. Even when you get to level 80, do not spend them, as you wont have nearly enough for what you need.
Once you reach level 80, there are several things you need to do. Firstly I would suggest not going to BG as you will get destroyed. Instead doing your HC's and getting PvE gear. Once you have decent PvE gear, you can start doing things Like VoA which allows you to gain your PvP gear. Also with decent PvE gear you can half decently play PvP BG, although with no resilience you are bound go get done.
Once you have accumulated the Honour Points you need, start buying your PvP gear. First of, you want to get the set bonus that gives you 100+ resilience, this is two items. If you want to do PvP you need to get this quickly, therefore you need to get the cheapest two items, (normally shoulders and hands if I remember correctly). Once this is done you are good to keep adding, once you have your 100+ resi, you'll find it much easier to do PvP but not that wrathful gear is out, our still going to struggle.
Additionally what you want to do is gain Sons of Hodir reputation to allow you to get the head enchant for when you get your head.
REMEMBER: Always gem and enchant your itmes with the best you can possibly get as soon as possible.
You can find detailed information on available mounts at WoWpedia. I'm not going to list all available mounts here, but you can find them all at that page. I'll try to summarize the general approaches you can take to getting a more exciting epic flying mount.
Basic wyverns are available from vendors when you hit level 70 and buy the Artisan (300) flying skill. Other mounts like the various drakes are typically available as faction mounts or rare drops.
Some are boss drops such as the Bronze Drake, others are reputation rewards such as the Netherwing Drake Mounts or the Red Drake. If you're a tailor, you can craft a flying carpet or its winterized version. Engineers and alchemists can also create flying mounts for themselves.
You can also buy this blue wind rider from Dalaran, but it's not terribly different from the common mounts except in colour.
Last but not least, there are mounts like the Violet Proto-drake that are awarded for getting certain achievements.
As you can see, the list of options is long and varied. Since you're nearing level 85, pretty much any of these should be available to you at least as far as plain level requirements go. Which mount you should go for first depends entirely on what you feel like getting and how much time/gold you're prepared to spend to do it.
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Once you're in Outlands, the gear earned from quests is much more prevalent, so the Heirloom gear gets less relevant at that point. That said, also keep in mind other characters you might play that could use the same heirloom gear.
As far as which piece to choose first, there are mainly two ways to go. The shoulders and chest are a popular choice overall due to the XP bonus built in. However, for a very gear dependent class like a Warrior or Rogue, an heirloom weapon would be a great help in the leveling process.