I'm going to start this by saying that it's perfectly normal to take your first steps in the world of bird photography using less expensive gear. In fact, I strongly recommend low- medium budget solutions for beginners. A less advanced camera/lens combo will demand more from you, the photographer. It will sharpen your senses and it will help you become more skillful a lot faster than good, easy to use gear.
But there comes a time in every amateur's life when he or she wants to get more serious about this hobby, maybe even turn it into a job. As much as I'm a fan of doing big things with small tools, I have to admit that professional results demand professional gear.
With that said, bird photography is one of the most demanding genres when it comes to equipment.
Let's start with the camera body. I'm going to mention a few recommended cameras, but you shouldn't limit your options to what I'm telling you. Knowing what specs to look for is far more important than going buying a camera after getting a recommendation. No matter what type of camera you go for, be it a DSLR, a mirrorless, a full frame or a crop sensor camera, you have to chose something fast.
If money's no object, definitely go for a pro body. Canon's flagship, the 1Dx is the choice of Canon pros. It's capable of an amazing 14 fps and with 61 AF points it will do well even in the toughest conditions. Nikon's D4s is considered the best option currently available. This is not me saying it, it's other pro wildlife shooters. The ISO performance of this thing is mindblowing and the speed is high enough to use it for shooting supersonic jets.
As second bodies, or options for smaller budgets for that matter, I would definitely recommend the 7D MK II and the D7200. These cameras are high speed, high performance bodies ready to track even the fastest birds, for a fraction of the cost of a pro body. They also have one very important advantage which is the crop factor. If you take a 600mm lens and mount it on a 1.6x crop camera, you get an astonishing 960mm lens!
Speaking of glass, you want to stay as far as possible from your subjects. Most birds are very suspicious of predators and it's very easy to be confused with one. Both Canon and Nikon produce astonishing 500mm and 600mm lenses. These are optical and engineering marvels that will take a serious toll on your budget, but they will get give you the best image quality available.
There are less expensive options, like 300mm lenses or even 70-200mm f/2.8 zooms that can have an extender mounted for extra reach. You won't get the same image quality as with a prime lens, but you'll still remain in the professional's league. As a final thought regarding lenses, try not to go under 300mm because that just means you'll have to get closer to the birds and risk scaring them away.
I don't recommend using flash for bird photography. I prefer natural light, but that's more of a personal choice. There are brilliant photographers out there who use flashes to take astonishing bird photos. I'm just not one of them.
Using a long lens and a heavy body is not something you can do handheld. Or at least not for more than a few minutes. I recommend getting a solid tripod that can handle the weight effortlessly. Vanguard, Sirui and Manfrotto all make good tripods for heavy loads.
If you don't want to carry that much extra weight, at least get a sturdy monopod. It will take the load off your hands and while still allowing you to be flexible.
A good binocular is highly recommended for locating birds. It will make it a lot easier to spot your subjects using one of those than having set up the massive lens every time.
Finally, be very careful about what you wear. Birds in their natural habitat are very suspicious and they scare very easily. Wear colors similar to those of the environment and move around slowly and quietly. Wear some tight waterproof boots if you can. A bird photographer's life often takes him or her to less than friendly environments.
If you want the best gear for bird photography as well as the best technical advice you can get, I suggest going to the 19th Annual Space Coast Birding and Wildlife Festival in Titusville, Florida in January 20-25 2016. It's one of the biggest worldwide events dedicated to bird photographers and nature lovers.