Babies, children and pets can be hard to photograph. They are generally darting around, they have very limited attention spans and can be difficult to direct and control. Here are some tips which should work well on almost any camera.
1. Find something that amuses them or attracts attention
It's always good idea to have a sound (cat noises or squeeks are quite amusing) or an object which kids find interesting. It's good to experiment with this (or ask the parents! :)) before actually shooting since getting the attention of the little one is definitely going to be key to getting shots of them where they are actually looking at the camera. Squeeky noises, arm waving and bunches of keys are almost universally loved.
2. Light: Soft and Plentiful
Soft light is easier to obtain than you think. If you are outdoors simply find a nice shady spot, if you are indoors, a room with big windows and sheer white curtains works very well. If you have neither of these, bouncing a flash will help immensely
3. Go Low
Always position the camera down to the eye level of the kid. This will make for a much more interesting perspective and it avoids them cranking their neck to look up at you. Going even lower can also make for interesting photos.
4. Pick your camera settings
I usually like to use a fast shutter speed and a very large aperture when shooting kids. Shutter speeds of 1/250s or faster will ensure that as the kid bounces around you will get a sharp photo. Aperture depends on how good the camera is at focusing: I use a Canon 5D mk3 and a Sony A6000, both of which have excellent autofocus systems. As such i'm comfortable using f2.0 to get a nice blurry background with the face in focus.
In cases where your autofocus system is less accurate, either select a smaller aperture (say f4.0) or focus the camera and keep it locked while waiting for the shot.
Kids require patience. They might ignore you for a long time and then give you a couple of seconds in which to take the shot. That means that you should have your composition, camera and focus all set so that when the opportunity presents itself, all you need to do is hit the shutter. Continuous drive isn't a bad idea at this point (unless you're using film :P)
6. Post Processing
This is a matter of taste, however I prefer keeping things simple at this stage. You might want to remove any pimples/mosquito bites using spot removal tools, and converting to black and white will focus the image your little one's expression.