The general rule of thumb for backgrounds is to make sure it is as simple as possible. Generally, you don't want the background competing with the subject. In 2D, elements of the foreground and background tend to merge visually, as noted where trees appear to growing out of heads. Look for backgrounds free from clutter.
Framing is the process of deciding where the edges of the photograph should be. What should and should not be in the frame? A quick rule of thumb is to fill the frame with your subject. Don't leave a lot of extra space around the sides. Also consider close crops on faces or detail shots. Don't be afraid to cut off a head or two if the picture calls for it.
Our eyes are drawn to the point of sharpest focus in a photograph. It's important to make sure that the camera focuses on a point of interest. A common mistake that camera auto-focus sensors make is to focus on the background. Be sure that your subject is in focus. For portraits, it's usually a good bet to focus on the eyes.
Shoot from a variety of angles. Get down on your knees, or your tummy. Climb up on stepladders. Do anything to see things from outside your usual eye level point-of-view.
Wide angle lenses tend to create perspective distortion that exaggerates the size of objects located around the edges of a photograph. Anything facing into the lens will be enlarged or elongated. Distant objects will appear more distant than they should. The distortion can be desirable or ugly, depending on the photograph. It is often undesirable in portraits, so keep it in mind when you're composing your photo. Step back a bit and zoom in, particularly if you're shooting a beauty portrait.