Be Prepared
Always have your camera out and at the ready.People generally do not like to have their photo taken so it's up to you to be more assertive and do it as quickly as possible.
Charge your batteries and/or get a second one.
Clear your memory cards and/or buy additional ones.
Composing:
Choose uncluttered areas to photograph groups.
Fill the frame with your subject
Keep in mind the relationship of your subject and their background. Be careful that people do not look as though things are growing out of their heads.
Use shallow depth of field to blur out background.
Use rules of thirds to compose photos, move your subject off center to one third or two thirds of the frame.
It is always better when shooting photos of people if possible, stand back and zoom in. The zoom lens makes people look better as opposed to shooting with a wider angle of view-similar to what a mobile phones camera will do . See related post.
Lighting:
Lighting is everything.
Turn more lights on.
Sometimes indoor photos look better without a flash and outdoor photos look better with a flash.
Use the red-eye feature to help eliminate red-eye.
Red eye happens because the flash is so close to the lens, using an external flash, if your camera takes one, is a great way to get the flash higher than the lens to eliminate red eye.
Using an external flash will also allow you to bounce a flash off the ceiling to get softer light.
Use your flash on a different setting like, fill flash, or slow sync flash to change the out put of the flash, both of these settings allow the camera to take in more ambient light that is in the room. This will give photos a warmer glow and not have harsh lighting that is overpowered by the flash.
Use the flash in the slow sync mode to take photos of Christmas trees, candles and holiday lights.
Use a tripod! Even a small point and shoot camera can take a tripod. This will allow you take take long exposures of holiday lights to get the best result.
Use a remote or the self timer:
Using the self timer on the camera is great for obviously getting in the photo. But can also be used for those long exposure photos to eliminate shake as when you press the shutter on a camera it will shake it a bit.
Better than a self timer is a remote for your camera, this allows you to get in the photos and take multiple photos with out having to get up and reset the camera.