Here is what you need for shooting kids:
1) Speed! Kids move quickly, their expressions are mercurial. A fast lens with a fast camera is a must. So, an SLR, but instead of a kit lens (which is not a 'fast' lens), you'll need something like the 50mm f/1.8, a prime that's ~$100 on the major systems. That lens is fast, meaning that it will find focus quickly and require less open shutter time (ie, take a picture faster), than a kit lens. I'd stop down to about f/2.5 or 2.8 in order to make sure to get the plane of focus wide enough to get everything.
2) Light! A flash, either an external or on the camera itself, will be something of a help, but the problem is two fold: the fastest shutter speed is generally 1/250th of a second (1/500th for an older Nikon D70), which may be about the speed you want to be shooting, or you may want to go even faster. The other problem is that flash can be harsh, and the resulting photos can have a look that you don't want. So, it's better to be outside or to have lots of light inside to shoot kids, and use a flash as a backup.
3) Distractions. You need to get the kid to look at you, so keys, shiny things, etc help for younger kids, or saying naughty words for older kids (whooo! You said something you shouldn't have!). When I was growing up, my dad used to say, "say shiiiii..." and we'd all have fits of laughter about that.
4) Be patient, but be fast. I find myself waiting for the shot as the kids are playing, waiting for that fleeting expression that I want to get. Once it's there, the camera has to be ready to go, no shutter lag or the like. That could be five minutes of waiting followed by a bunch of shots in a row.
Also, @ahockley's composition suggestions are good. Don't shoot from your head height, shoot from theirs (or, to really mess with perspective, shoot from the ground up, to give them the Godzilla look. Toddlers in particular like to think that they are huge, apparently).