Flies require TLC to be made correctly. If they don't get good attention, they'll often fall apart upon first use. Big Y Fly Co. offers almost any fly you can think of, at an affordable price.
You really don't want to cheap out on flies. Such fine, detail-oriented contraptions require a meticulous hand. If you opt for cheap, factory-made flies, you'll find that they often fall apart on the first use. Collectively, cheap, lightweight thread, skimpy portions of cement or glue, and little to no oversight don't bode well. Suffice it to say it's a good idea to stay away from Walmart, Dick's, or Cabela's for flies. I've found all this out the hard way, and while there might be exceptions, you'll put yourself through a good deal of hard-earned spending and learning.
The best way to get your flies is to go down to your local fly shop if you have one. But, if you already know what you need, or local prices are too prohibitive, there is one online retailer that I trust: Big Y Fly Co.
Pro tip: A good trick when buying (or tying) any flies is to apply an extra dab (or coating, depending on the fly) of head cement, or even just clear nail polish or nail hardener. Flies can take a thrashing, both from fish and, say, rocks and trees (the latter depends on how good of a caster you are).
Like with any lure (but maybe especially with flies), wash them with a generous helping of fresh water after use, especially if you've been fishing in salt water, and your lures will live much longer. Salt is bad for any kind of hook, but it dries out hair and feathers, and they'll become brittle and start to break off.
There is a cacophony of views on the matter over at Reddit (here and here), and one expert at Field & Stream commends Big Fly Co. when discussing the pros and cons of buying flies online.
Ultimately, you should learn to tie your own flies. It will save you money over time, but it will also teach you more than you could possibly ever learn otherwise about the fish you pursue. Orvis makes a kit that is very good and reasonably priced.
Pros: Affordable, wide selection
Cons: Only that you'll have to wait for delivery
Shop all flies at Big Y Fly Co.
Buy Fly Tying Starter Kit from Orvis for $158