In Leonardo da Vinci's famous Mona Lisa, his female subject is seated in front of a simple Italian landscape. There's a bridge, a river and a few mountain peaks -- it's nice, but it doesn't attract attention away from the Mona Lisa herself.
Just like Leonardo, keep the backgrounds of your family pictures simple. Your goal is to focus attention on your human subjects, and if you're photographing them in front of a switched-on television or a cluttered coffee table, the focus will be lost. What you'll want to look for is a neutral background: Something that's not too flashy and won't overpower your subject. Outdoor settings are cheap and easy to light, but be careful that you don't put your family in front of trees or poles -- they'll look like they're growing out of the person's head. In any outdoor setting, scout around for the backdrop that's least likely to stand out. If your family is at the beach, photograph them with their back to the ocean rather than the parking lot.
The weather's a little rough? If you live in a city, you're probably surrounded with free indoor backgrounds. Pack the family into a minivan and drive them to a nearby university atrium, library or hotel lobby. Many portrait photographers also choose to photograph subjects in their natural environment -- such as the office, home or laboratory. If your son plays soccer, photograph him on a soccer pitch. If your wife loves gardening, get her posing in the tomato patch.
In all cases, try adjusting your camera lens so the backdrop is out of focus. With a blurry background, your subject is instantly made to leap out of the scene.