So here's a few tips I've picked up along the way. Even if you're not leveling, maybe some of these will help. And I'm sure there are plenty of great ways to make sure your bags are clear that I haven't discovered yet (I know for a fact that there are plenty of addons out there that I haven't had an interest in yet) -- feel free to share your own tips in the comments below.
1. Sell the junk (even if you think it's not junk). The quickest way to clear out some inventory is to run to a vendor and just start selling. It can be a tough decision sometimes in both ways: I tend to want to keep things because I have a base packrat tendency as it is, and I always know there's a use somewhere for even the cheapest junk: maybe some Shaman will want that Fish Oil! But the fact is that you have to be kind of ruthless when clearing out the bags -- sell the grays (there are addons that will help with this, though I haven't used any), sell the whites unless you know there's a market for them, and I often even sell the greens -- while die-hard AHers will say you can always make more than vendor for AH on green items, I'm just not patient enough to keep posting (along with the fee that goes with it) until the items sell for a gold or two more than they would for a vendor. Yes, maybe that's why I've never been able to buy a Traveler's Tundra mount, but especially in these days of level 80 dailies and the gold you can earn from them, the bag space is worth more to me than the pennies I'd earn otherwise. If you really can't bear to sell things to a vendor, then...
2. Get a bank alt, and use it for everything. Bank alts are awesome for freeing up bag space. My Paladin is a Miner and an Engineer, and all the ore and Engie doodads that I collect go straight back to my bank alt until I need them again. Given that mail between alts is almost instant and super cheap, there's no reason you can't have a dedicated alt standing by the mailbox in Ironforge or Org with a bank of its own running all of your main's equipment. If you really want to go all out, you can use a character that's already at a relatively high level, and powerlevel their enchanting high enough that they can disenchant all of those greens and make some real money. Just make sure to deck them out nice so everyone knows what they're there for.
3. Gear up right. The number one investment you can make in your character, no matter what class or spec or level you are, is more bag space. Buying bigger bags is always worth it -- if you can fill up that one extra bag slot with something you can see for a few g on the Auction House, your bag will pay for itself within a few hours of leveling or grinding. And don't forget profession bags -- while they have some weird requirements (why don't jewels fit into my Mammoth Mining Bag again?), they're almost always worth it, especially while leveling or grinding for a profession (and when you're done with them, put them in your bank and have them hold profession stuff there). We're also lucky, because we're playing in the time of the Gnomish Army Knife -- if you can afford one or if you have an alt or guildie who's willing to make one for you, pick it up early and use it often. On my Miner/Engineer, that thing saves five slots of my bag space.
4. Organize! Everyone has their bags set up different. Unfortunately, Blizzard still hasn't implemented my bag naming idea yet, but you know you how you like your inventory set up -- I like to keep oft-used permanent items like tools and food in my backpack, extra and off-spec gear in my second bag, crafting mats in my third, and empty space out from there. Everyone has their own scheme for organizing, but as long as you have a scheme that works, stick to it, and you'll be able to not only find things when you need them (I keep my potions on the action bars, but occassionally I'll forget to drag the healthstone over and have to dive into my bags to grab it before I'm killed during a raid), but you'll know when things are out of place that need to go elsewhere.
5. Regular maintenance is better than emergency cleaning. This is one I'm still bad on -- in fact, I know I've forgotten to fly back and train up for the past few even levels I've hit. If you stop by a vendor and clean things up every time you're in town, then you'll be better off when you do start picking up lots of junk. Keep an eye out for quest items, especially -- I tend to turn in quests that I picked up a while ago, leaving one weird spot in a strange place in my bag that then gets filled up with vendor trash that I miss over time. Addons like Onebag can help with this as well, but really, the more organized your bags are and the more time you take to make sure they're in order, the less chance you'll end up having full bags when you really need to pick up that quest item.