Finding the perfect combination of items for your toon isn't easy. You have to use wowhead, the model viewer, or some other tool to browse through hundreds of possibilities, determine if you can actually get the items and where they can be found.
Best case scenario you can craft the item you want, buy it outright from a vendor, solo some old dungeon boss a few times till it drops, or do a few quests to earn it as a reward.
Of course not all items are that easy to get, you'll undoubtedly find some items that drop in raids. Even if you can solo the raid you have to wait a week between attempts. More than likely you'll need to find other people interested in running the raid with you, and that isn't always easy.
Then the absolute worst case scenario are items that are just world or trash drops. Those greens that have a one in a hundred chance of dropping off of mobs of a certain level. I want three of those for my shaman right now and I must say AOE clearing an old vanilla dungeon repeatedly is quite boring.
Of course there is always the auction house, but these hard to find greens are just as rare on the auction house and it certainly isn't worth buying an item on a different realm and spending 25$ to realm transfer it to yourself. I did start to consider it until I realized I had all 10 slots filled.
So I check the auction house every time I am nearby now, and I farm for a while when I can stomach it. It seems the most efficient way to go about transmogrification is to come up with the items you want on as many toons as possible in hopes that some of the farming can be done in parallel. Still I've been doing one at a time, it takes a couple hours to come up with a good looking set.