Do you have the concept of the Rule of Thirds firmly in mind? That's good, because there are several other basic compositional factors that can also be followed or ignored as the photographer sees fit. They should at least be considered. Knowing what they are will provide you with a set of options.
1. Isolate the center of interest from other elements in the scene. Lines running through the main subject, either in front of or behind it, tend to diminish its impact. Especially distracting is an object such as a telephone pole that appears to be growing from a person's head. This type of compositional goof can be avoided by careful examination of the background as you hold the camera up to your eye.
2. Shoot at an angle. If you are not satisfied with the way it looks head-on, move around or have the subject turn so that you are shooting at an angle of approximately 45 degrees, or whatever angle looks best. This will add interest to the photograph as well as enhance depth perspective - the illusion of three-dimensional space in a two-dimensional image. Converging lines and the diminished size of distant objects are two factors that add to this feeling.
3. Fill the frame. The generalization is, include only what is important for the concept you wish to convey. This is a touchy one. Is the most interesting feature a building and its immediate surroundings, the building alone, or is it a unique window or door? Chances are, each of these would result in an interesting photograph. Don't limit yourself. Take a variety of shots from various distances, keeping in mind the fundamentals of a good photograph with each shot.
4. Give the camera a quarter turn when appropriate. Not all subjects lend themselves to horizontal composition. Generally, the format of the subject determines the best composition for the picture. To show the stateliness of a tall tree or building you would want to use vertical composition. Most landscape scenes and athletic events are predominantly horizontal. To determine the best format, horizontal or vertical, select the most important element in the scene and use that as your guide. If you're undecided, take horizontal and vertical shots. Either or both might give you just what you're looking for.
5. Be aware of the focal point. The focal point of a picture is the first thing that catches the eye of the viewer. It could be the center of interest, but not necessarily. It might be a brightly colored object, something of unusual size or shape, or something unexpected. Whatever it is, the focal point should be supplementary, if it's not the subject itself, and not compete with the primary subject. With the camera up to your eye, look for an object, or maybe a ray of sunlight, that demands your attention. Is it distracting, or does it add to the mood of the picture? If it distracts from the center of interest, try to remove it, either physically, or by your moving around or changing your angle of view.
Remember, composition is a conscious placement of the subject matter as you see it through the viewfinder. Now that you know the rules of composition, you can use them or break them as you see fit. Either way, it should be a conscious decision.