I've got two nieces (2 years old and 6 months old) that I have photographed, dare I say, ad nauseum, and here are some things I've picked up along the way.
- number one, hands down, is "watch the backgrounds" but I think Derrel has sufficiently driven that point home already. DAMN, is it frustrating when you get a nice sharp shot of an awesome moment or facial expression, but it looks like the child is growing out of some background adult's crotch, or they have a lamp cord sticking out of their ear, or there is a child's car seat and overflowing diaper bag in the background, or..... you get the idea.
- less-than-perfectly-clean faces aren't really a concern for me usually, but that's up to you to decide what you'll tolerate. It's worth noting that minor cheek smudges, etc. can easily be corrected in post; much more easily that crappy backgrounds can be corrected, anyway. Watch for crusty boogers, though... yuck.
- if the weather is nice, get outside. There are all kinds of advantages to this over indoor shooting--but be wary of really bright sun, as it will make the kids squint and have harsh shadows on their faces, neither of which look good. To make matters worse, you have to watch closely to see the problem on your LCD viewfinder, or you might not catch it until you're downloading your pics to the PC and it's too late. If mid-day sun is your only option, there are things you can do to manage it (use shade, use modifiers, etc.)
- re: "use a fast lens". Obviously, that is a desirable thing to have, but the reality is, aside from the ~$100 "nifty fifty" f/1.8 prime lenses, most lenses that could reasonably be called "fast" are pretty expensive and therefore the vast majority of us mere mortal hobbyists and novices don't have them in our camera bags. As far as the nifty-fifty goes, it's a nice lens for the money, but IMO it's a horrible lens to be using on a crop-sensor camera indoors in this situation. At an effective focal length of 50 x 1.6 = 80mm (Canon's crop factor; I'm sure Nikon's is similar), the field of view is too narrow to be useful in cramped quarters, ESPECIALLY when the subject matter consists of kids who can't sit still for 2 seconds. Large, open rooms and back yards are better situations for the fifty if you must use it, but honestly, with the modest kit I have, if I'm indoors I'll go for the 18-55mm kit lens + bounce flash all day long over the fifty.
- always be mindful of shutter speeds. Kids tend to make unpredictable "snap" motions which can cause motion blur in their limbs, etc. and ruin an otherwise perfect image. Yet another advantage of shooting outdoors with abundant light.
- longer zoom lenses are awesome for taking pictures of kids outside. I like to use my 55-250, sit off to the side of the yard, and snap candid photos of the kids. The long focal length usually provides a nice softening of the background, and you can just unobtrusively snap all the photos you want of the kids being themselves.
- I'm sure some will disagree with this suggestion, but don't be afraid to leave a tad of extra space around the subject so that you can "customize" the final crop in post. The more you can get right "in camera" the better, but when trying to capture such dynamic moving subjects, nailing the exposure and the focus are more important than the framing, because you can crop later. I'm always staggered at how much impact and visual appeal can be added to some photos by doing this.