If I had to guess, I would say that 99 out of 100 hobby photographers never bend their legs. It's not so hard to see why - human beings view the world mostly from one or two perspectives: standing up and sitting down. Occasionally we will also lie down in a place other than our beds or the sofa, but for the most part everything we see comes to us at a perspective of somewhere between five and six feet off the ground.
So most photographers don't think about finding other angles, because the angle from which we view the world most of the time is so familiar and comfortable. But the sad truth is, it's also boring. When you walk past that favorite city landmark and snap a photo of it, the chances are really good that your photo will look exactly like the last thousand photos that the last thousand photographers took of that same landmark--unless you spent some time thinking about your subject and how you could capture it in a unique way.
First the obvious: Change your Angle
Anyone who photographs children professionally will tell you that the best way to get an engaging photo of a child is to get down to his or her level. When you see a child as a child sees the rest of the world, it adds a whole level of depth and meaning to an image that would simply not be there if you photographed that child from your normal adult perspective. The same goes for everything you photograph, and I mean everything! A flower will be infinitely more interesting shot from ground level looking up than it would ever be shot from where you happen to standing. A cafe filled with neat rows of tables will look far more intriguing photographed from several stories up, looking straight down, then it will be shot from street level. And yes, you will have to learn to get over the fact that you look stupid lying on your back on a carnival walkway or eye to eye with a dachshund. Your camera excuses you. And if there are those who don't think it does, well, they obviously don't have an eye for fine art anyway.
Now the not-so-obvious
By all means, make sure you get a photo of the Washington Monument. After all, you want to have some sort of record that you were there. But remember, that straight-on shot of the Washington Monument is just that - a record. It isn't really art, and probably won't be something you'd hang on your wall. So after you take that straight-on shot of the Washington Monument, wait around and see what other photos you can grab. Capture an image of a little boy with his head tipped back, staring in wonder at the height of the monument. Then get a couple of shots of that other kid who is bored to death standing in line. Now angle your camera in such a way that the flags nearby are interacting with the monument in an interesting way. Or back up even more and see if you can use Washington D.C.'s famous cherry blossoms as a natural frame.
Play around with camera equipment
Use a fish-eye lens and you'll automatically have an unusual angle, even if you aren't necessarily taking the photo from an unusual angle. Fish-eye lenses create distortion, which lends a surreal feeling to anything you're trying to photograph, from a person to a landmark. You can also experiment with a neutral density filter or a circular polarizer to get a dramatic sky, or to capture motion blur during the day. Messing around with your camera's white balance setting will also add a hint of uniqueness to your subject by creating a color cast that can dramatically affect the mood of your image, or just make it look surreal.
If you're just out for a day of fun and photographic experimentation, bring along a smart phone or an Android-powered camera and use one or more of the many photography apps out there that will allow you to apply interesting effects and filters to your images. These filters can give your photos anything from a vintage feeling to a space-age one. You can also go one step further and buy any number of lenses or lens attachments for your DSLR that will help you do pretty much the same thing--from pinhole attachments to tilt-shift lenses and other gadgets like the Lens Baby or Holga lenses.
Break some rules and see what happens
Try shooting with a strong backlight; this will give you interesting silhouettes or images with a halo effect, and can also add lens flare and other artifacts that are generally undesirable but under the right circumstances can add an unusual and interesting element to your photos. Or you can try hand-holding your camera at a slow shutter speed, or adding a little motion blur to a landscape image with the help of a tripod and a neutral density filter. Blur can sometimes be interesting and even camera shake can add some intrigue - under the right circumstances.
You can also try breaking some compositional rules. Do you hate that confining "rule of thirds?" See if you can get a good image by framing your subject dead center. Hint: balance is the key to making this kind of photo work. Try to balance your subject with his, her or its surroundings to achieve the most success in any image that is a blatant violation of this rule.
Experiment, experiment, experiment - both in camera and off
You can always give your image that extra little tweak in post processing to make it really unusual. Try adding a splash of color to a black and white image or dramatically darkening the sky. Apply some subtle filters (the key is subtle - you don't necessarily want your image to scream "Photoshopped!") and see what kind of results you get. Experimenting in this way can also be useful for helping you develop an eye for unusual images while out in the field.
And when you are out in that field, always be on the lookout for ways in which you can capture an unusual photograph. If you start consciously shooting this way, eventually your efforts will become subconscious, which will in turn make you a better photographer.
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