I'm not going to kid you, photographing these little quicksilver characters is a challenge. They're here one second, then gone the next. They can be surprisingly tolerant of your presence, but then flush at the slightest movement. They're tiny, and they always seem to be just out of reach of whatever lens you have on your camera. Having success with these reluctant subjects requires knowledge, stealthiness, persistence, and most of all patience. You will throw away a lot of photos for every keeper; my success rate is hovering at around 1%.
All that said, when you do manage a keeper, it can be magical. So many things have to come together to make a successful bird photo that it feels as if you've really accomplished something when it happens. Following are a few things I've learned so far.
Every camera sensor produces a defined number of pixels, from around 12 million on the low side (phone cameras, point-and-shoot cameras, and older DSLRs), to upwards of 50 million on the high side (professional DSLRs and medium format cameras). The higher the pixel count, the higher the detail captured by the sensor. As bird photographers, our goal is to put as many high quality pixels on our subject as possible. All other things being equal, the more "pixels on the bird" (as Kenn Threed puts it), the more detail is apparent in the photograph.
The obvious way to get more pixels on the bird is to purchase a high pixel-count camera with a high powered telephoto lens. This is the approach taken by most professionals. This type of gear will provide the highest success rate while also producing the highest quality output. The downside is that this type of gear can be exorbitantly expensive (an outfit may run upwards of $10,000) while also being large, heavy, and difficult to work with.
Some amateur birders use what are called "superzoom" cameras to get more pixels on the bird. These are entry-level cameras with fixed lenses that have incredibly wide zoom ranges. The Canon SX60 is a popular model that boasts a zoom range of 65X, going all the way from 24mm-1365mm. This camera retails for around $450, yet is has a lens that reaches out to extreme telephoto distances, similar to professional lenses that cost upwards of 10 times the price. It manages this by using a tiny sensor which, due to the physics involved, allows for a small (inexpensive) lens with long reach. The downside is that although great detail can be captured with these cameras, the overall image quality is relatively low. Image noise at anything other than base ISO can be pronounced, the overall pixel count is low which limits the potential for cropping, and the focus and image processing systems are substandard when compared to professional and even semi-pro cameras.
I currently operate in the no man's land between the superzoom and professional levels. My birding setup consists of a Fuji X-Pro2 and a Fuji XF90mm f/2 lens. The 90mm Fuji is a stellar lens, one of the sharpest ever produced by any company in any era, but it's not what anyone in their right mind would call a birding lens. On my camera body it's the equivalent of a 135mm portrait lens. Most birders prefer lenses in the 400mm-600mm range or even longer. Fortunately, I'm only photographing birds in my backyard, so with some careful technique and a few technical tricks, I've been able to have at least some success using tools not quite right for the job.
Where my gear is lacking, I've been able to make up for it by getting my camera closer to the birds (that "more pixels on the bird" thing again). Through experimentation, I figured out roughly the best distance for photographing birds around our feeder station, then I moved our backyard bench directly in front of the feeders at that distance. When I slip out into the backyard with the camera, I very slowly move to the bench and sit down. The birds always flush, but if I'm careful and deliberate, they usually only move a short distance away. I then sit perfectly still on the bench, and often they'll come back to feeding within 5-10 minutes. If they're very active, I can sometimes even stand up and move around the tree to experiment with different compositions. The amount of movement they'll tolerate varies greatly depending upon how aggressively they're feeding (the more birds and the more aggressive the feeding, the more tolerant they are of me).
I also have a technological solution that works well for a particular type of photo. I have a wireless shutter release that can trigger the shutter from up to 350 feet away. This allows me to set the camera on a tripod very close to the feeder, then trigger the camera from inside the house. The obvious disadvantage to this method is that the camera must be positioned and pre-focused on one point only (usually right on the feeder), which doesn't allow for any creativity in composing the image. While this seems like a nice solution, my best shots have all been made hand-held, moving into different positions very close in to the birds.
When all else fails, there is always cropping. This is where having a high pixel count sensor and a sharp lens comes into play. The higher the pixel count, and the sharper the lens, the higher the quantity and quality of the pixels on the bird. This is where the quality of my Fuji system provides some advantage. Even though it's not a pro-level birding outfit, the image quality at the pixel level is very good, which allows for fairly extreme cropping while still maintaining a good amount of detail and clarity.
Putting it Together
Using a combination of a reasonably long lens (I'd say at least 135mm equivalent) on a relatively high pixel-count body (I'd say at least 24 megapixels), stealthy maneuvering, and deep cropping can result in surprisingly good results. Working in this manner naturally results in a lot of missed shots (mostly due to the necessity of being so close to the birds), but with a little practice, patience, and good technique, it's possible to produce very satisfying images of backyard birds without taking out a second mortgage on your house.