1. Remove the clutter. Take a moment to pick up the clothes off the floor or put some toys away. Move the trash cans and yard tools. These things will create a distraction and take the focus off of the subject. Sometimes this simply cannot be done without missing a moment. Don't worry too much about it but clean up the background when you can. You can also use framing and composition to remove clutter with your camera. This leads to tip #2.
2. Get closer. However close you are, half the distance. What are you photographing anyway? Not the background or the 20 other kids on the playground. Get close to your subject. This allows you capture emotion that you won't see from a distance. It eliminates distractions and clutter and focuses on what's most important. You'll want to remember details like freckles and kool aid mustaches.
3. Change perspective. We all walk around looking at things at eye level every day. It gets boring. Imagine what things must look like from the height of your children. Much more interesting I'm sure. Get down on their level and shoot up. See it how they see it. Hand them the camera and let them take the shot. See yourself how they see you. Or get higher. Climb a ladder or stand on a rock. Having your subject looking up gives a different look altogether. Grown ups love this perspective because it is very slimming. Just change it up.
4. Find the good light. If you are not planning and posing a shot you need the light to be in your favor. Outside snapshots are pretty easy. Just find spots that allow your subject to look at you without having to look into the sun. The shade of a tree or the side of a structure are great options. Or put the sun behind them. Inside may prove a little trickier. If you want to avoid using a flash, and trust me you do, keep the windows open. Raise the blinds, pull back the curtains, open the shutters. Let in as much light as possible.
Just a few small changes and bit more attention to details can make great photographs out of everyday snapshots. Just make sure you are taking them. Don't wait until you think you have the technique down. You'll learn as you go and probably have some fantastic happy accidents along the way.