To get Azeroth flying, you need to visit a flying trainer anywhere. Cost is 250g.
You'll want to start your 80s in either Mt Hyjal or Vashj'ir. I think you can get to either by flying, but you will have portals to either available in Org & Stormwind. Though you may have to do pre-quests to use the portals. If so, I don't currently know what they are.
For Orgrimmar, your flight trainer is by the flight master -- take an elevator up in any corner of the Valley of Strength and it'll dump you near the flight master. Run to him and the trainer is beside him. Once you have flying, go north across the bridges until you get to the portals in the Valley of Wisdom (the platform is shown on your map and looks like a small grey Fortress of Solitude).
For Stormwind, your flight trainer is also by the flight master. The portals to Mt Hyjal and Vashj'ir are northwest of Stormwind Keep.
Remember to empty your bags, get your potions and buff food in hand, recheck your talents, repair your gear, make sure you've learned all your skills, and sit on a flying trainer so you can start a new zone with no (okay, less) poking around in confusion once Cata's out.
And remember: rested is better than any other leveling buff. It quite literally doubles experience gained. For those of us who have fulltime jobs (woo, money!), rested is a big deal. If you do nothing else tonight, get yourself to an inn.
Good luck, and have fun everyone! Europeans who already have Cata, it's cool, we hate you. Enjoy Blizz loving you more than us today.