Fly rod, reel and line:
When it is time to purchase a fly rod and other necessary gear, there are some very important things to know and it can be very confusing for a beginner. The most important and necessary equipment is a properly setup rod, reel and fly line. I mention all three of these things together because all three need to work together...in fact, they need to be "matched". When I say "match", I do not mean color coordinated. I mean that the weight (wt) of the rod (typically 2 wt. to 14 wt.) must closely match that of the fly line (also typically 2 wt. to 14 wt.). The reel too has a weight (wt) associated with it (typically 4-5 or 7-8).
I will not get into the details of weight as it applies to fly fishing as it can be very confusing for a beginner. Just know for now that "matching" the "weight" (wt.) of the rod, fly line and reel are very important...not just for the beginner, for all fly fishers. Matching the weight of the rod and fly line is most crucial. As you will learn when starting to cast, if the weight of the rod does not closely match that of the fly line, it becomes more difficult to get the fly rod to "load". You can think of "loading" as making the fly rod bend during the casting stroke. All fly rods are designed to properly load when the rod weight closely matches that of the fly line. If they do not closely match, fly casting is much more difficult. In fly casting, you always want the rod to do as much of the "work" as possible. When a fly rod "loads" (bends), it stores energy that is released as the rod straightens and this energy is your friend.
As I mentioned above, fly rods and lines are rated from 2 w.t to 14 wt. The smaller weights are for smaller streams and smaller fish and the larger weights are for bigger water and fish. A 12 weight is a typical rod for fish like tarpon and other fish that weigh upwards of 50 pounds.
So, what should you buy? Most knowledgeable fly fishers would recommend a 5 or 6 wt. rod for a beginner...I agree. This is the most common size for most fishing situations for trout, bass and panfish up to several pounds. As you start to look around to buy equipment, you will quickly see that it is possible...actually easy to spend over $2,000 for just a rod, reel and fly line. Happily, that is not necessary although I do recommend that you purchase the highest quality you can afford. My first recommendation if cost is an issue is to purchase a combo starter kit with a rod, reel, fly line with backing from a company that really knows fly fishing. This will virtually guarantee you that the rod is properly matched with the fly line and reel and ready to cast. A few of the sources I can name with some confidence are Orvis (Encounter Outfit $169), Reddington (Crosswater Combo $169), Temple Fork Outfitters (TFO NXT Black Label Kits $219), Echo (Base Fly Rod Kit $160). I know there are other less expensive choices and a google search will find them. I am hesitant to recommend them as I know nothing about these companies, their background or lack of background in fly fishing, etc. The really inexpensive ones are made in China ande quality can always be an issue. Also, try to find one that comes with some sort of warranty.
Links to more detailed information
Waders and wading boots - There are many ways to save a little money as you move forward...waders and boots is not one of them in my opinion. Obviously, there are places and times of the year that waders may be optional but if you need them, again, buy the best quality you can afford from a reputable manufacturer and make sure they have a decent warranty. Being cold and wet will ruin even the best day of fishing! More information...
Nets - I mention nets for only one major reason. It is "Catch and Release" fishing. It is my personal opinion and that of many other fly fishers that releasing a fish, especially a "wild" or native trout is the right and best thing to do. In many parts of our country, wild trout populations are on a serious decline and, in some places, are difficult to even find. A single released female trout can spawn and thereby create thousands of new trout that can be caught for years to come. If you just intend to catch dinner, a net is just a nicety. A proper net has rubberized netting materia land is soft to the touch.
Fishing vest or other "pack" - As you move forward, you will quickly realize how much small stuff you will need/want. Having a convenient place to keep all of it handy and organized will be very useful. Here is a good place to save a few bucks as quality and warranties is not nearly so important.
Flies - They are flies...not bait or lures. Here, you should wait until you are ready to actually go fishing before you spend even a dollar. Although they are critical in fly fishing, there are literally thousands of fly "patterns" that primarily imitate insects (aquatic and terrestrial) and small baitfish. As you already know, different fish (species) eat different food and even the same species will eat different food as things such as , water conditions and even geographical areas change. A purchased fly will cost between $1 and $5 with the typical price being about $2-$3 so it is easy to spend serious money quickly. Talk to fly fishers and fly shops familiar with the water you plan to fish before buying flies. Over time, it is common that many fly fishers learn to tie their own flies. This can easily save over 50% over purchased flies. See "beginner fly tying" here...