Stick with the blue
While the point of leveling alt characters is that you don't need to rush to endgame, it's still an awesome feeling to fly through the levels. After all, if you're bringing eight characters up through the levels, then you have 640 levels to go. That's plenty of levels to keep you occupied, so you might as well use them up.
The best way to get the more return on your play time is to make sure you're always questing when your experience bar is blue. During blue experience, of course, you're gaining double the normal XP from killing monsters. It's not night-or-day experience levels, of course, but it sure as heck makes things go faster.
Prepare for money sinks
There are money sinks built into the game. Bags, mounts, training and bank slots are all examples of money sinks. They're all things you can get by without, of course, but they sure as heck improve your game experience. I can't imagine trying to run too far around Azeroth without a decent mount.
The downside of a series of alts is that you have to repeat most of these money sinks for each character. You need to prepare yourself for buying mounts and such over and over again. Be careful not to throw away any gray vendor trash, for example, because that's as good as money in your pocket.
Auction anything that you don't need, but never buy from the auction house if you can help it. Unless you've spent some time preparing yourself for the auction game, you're likely to just end up spending cash. If you find yourself needing a particular drop or reagent, check out how to get it on Wowhead and go farm it yourself.
Diversify your portfolio
Make sure you have at least one character who is learning each profession, so that you can create any of the most vital items along the way. This is especially true of tailoring and bags, since all your characters will need a full complement of bags as they progress through the world.
Other crafting items will be helpful to your army of alts, however. Blacksmiths can provide plate armor and weapons, leatherworkers will make leather gear. You'll find yourself occasionally using crafted gear for your characters, as you play a complicated game of "what's better for my character just now."
Try out new zones
If you have multiple characters, try and be sure to experience every zone in the game. Wowhead has a pretty good list of zones by level, and you can use it to try different paths for your character. Taking all of your Alliance through Westfall, for example, might mean you'd miss out on Loch Modan.
The World of Warcraft is a pretty big place and there's a lot to see. There's going to be even more new stuff after Deathwing rocks the planet in Cataclysm. If you're going to be leveling multiple characters, you should let yourself see it all.
Track your achievements
One of the nice things about having a single, main character at max level, with epic gear, and all the achievements, and killer titles is that you get to stand around in Dalaran to show off. But it's hard to do that as an alt player, because each one of your individual characters doesn't seem as impressive. But, of course, if you were to round up a half dozen characters who could all show off their collective awesome, then you'd have a much more visible representation of your work.
There are a couple ways for you to link your characters together to show off your achievements. First, if your alts are all named in some similar way, then other players will associate your characters together in their minds. For example, I might name my characters Grayfields, Grayskies, Grayrocks and Graytrees. Sooner or later, other players will get that they're all together.
The other option would be to create a vanity page for your army of characters. Therein, you can detail the information about your alternate characters. The web design of such a thing could be fun and give you the option to show off some fun screenshots.
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 pulling together enough cash for mid-level expenses such as mounts and dual specialization, to what to do when you finally hit level 80.