When I worked in a fly shop we had a saying "There are flies that catch fish and there are flies that catch fishermen." Fishing equipment is like jewelry to some; you're looking at so many sparkly shiny things and our heads start spinning with all the gear we want.
The sport of fly fishing doesn't have to be expensive, or really require that much that you need to take on the stream. Every fly fisherman needs a rod and reel, fishing line, leaders, and tippet material. Flies are up to you, but when it comes down to it I basically use about a dozen or so type of fly patterns on a regular basis when I go trout fishing, so I really make sure I'm stocked on my productive patterns. Of course I always have my forceps and nippers used to cut line with me, too. Other than that I don't need a lot of fly fishing equipment before I'm ready to hit the water.
You will obviously have a few other things like dry fly floatant if you are dry fly fishing, or strike indicators and weight for using a nymph. A hat and pair of polarized sunglasses to cut down the glare on the surface of the water are essential for me, but if I want a wading staff I grab a stick. I'm usually in the forest and they are everywhere. I have a fly fishing vest if I want to carry everything, but a very small sling pack can hold a few boxes and a water bottle. You can find all kinds of packs and vests that come in a variety of sizes to fit a billion things. I only bring a net if I want to take pictures or with clients, otherwise I minimize the damage to the fish by leaving it beneath the surface of the water and a simple turn of my forceps removes the hook.
Waders and boots are another range of expenses and should be reasonable to you and your regular fishing situations. Waders keep you dry as long as you don't expect to walk straight through stickerbushes. Anything with holes, expensive or not, aren't waterproof. Boots are good for not slipping. If you fish where you pull up to the stream next to the road then super-light wading boots made for hiking aren't for you. If you walk miles in, felt is going to be destroyed after a season. Full chest waders can always be rolled down, but if all the streams you fish are only three feet deep, then hippers are all you need. Insulated waders are great if you are fishing in cold Alaska, but you can just wear more clothes underneath to stay warmer. You aren't going to want those neoprenes in the middle of the Summer, and never necessary in the Shenandoah Valley of Virginia.
When it comes to essential equipment I advise the you to consider where you are going to be able to fish most of your time and go from there. You'll get more out of your money, though it is hard to go to an local fly toy store and not want everything on the shelf for your next fly fishing trip.
Fly Fishing Gear List You Need to get Started:
Fly Rod
Fly Reel with Fly Line
Leaders & Tippet
A fly box for holding flies
Appropriate footwear to wade
Hemostats/Pliers/Forceps
Line cutters/nippers
Not Essential Fly Fishing Gear (but nice to have)
A hat and polarized sunglasses
Fly fishing net
Rain Jacket
Vest or Bag
Split Shot