Your situation sounds a lot like mine.
As you say, there are a lot of good cameras out there.
One of the things you will probably be up against is that a lot of cameras come with the standard 18-55mm lens. It gets you started, and you will probably find the 18mm nice and wide, but you will probably find the 55mm end a bit short.
You can get another lens like the 55-200mm and switch between them. But then that means switching a lot when you want to be at about 55mm.
One option is to not get the 18-55mm lens and get a 18-70mm or 18-105mm lens as the kit lens instead. It will cost a bit more, but then you will have more reach with it. This might be able to postpone getting the 55-200mm (or 70-300mm) lens. But when you get the other lens, then it may mean less switching between them.
Of course, I say this and I went ahead and got the 18-55mm lens with my Nikon D5100, but that is because I'm planning to get a couple of faster lenses so I wanted to minimize money on the kit lens.
You mention taking pictures of kids.
One of the nice things I like about the Nikon is the built-in autofocus assist light. I find it works really well at letting the camera focus on my kids as they are running around at get togethers, where lighting is usually bad, especially with regular lenses. The other camera that has that is the Pentax K-r, which is the other camera that I want.
As for shooting indoor cheerleading and dance, it may still be hard. It's difficult because of the lack of good lighting. I am getting some faster lenses to try to deal with this, but that's about $1200-1300 worth of lenses I'm planning to get. The Nikon D5100 and D7000 both go up to ISO 25600, which may make it possible to shoot with regular lenses, although with some digital noise. Of course you can back off the ISO a bit and have a bit more motion blur. You may want to look a bit more into this. My general thought on noise is, yes it may be a bit noisey, but if it means getting the shot vs. not getting the shot, its probably worth it.
[EDIT: Oh, yeah. I was originally looking at the Nikon D7000, but when I looked at everything I wanted, I realized I didn't have the budget for the D7000 this time around. I wanted to spread the budget around to different things, so decided I'd go for the D5100. In the end, fitting everything into my budget was what made me decide the D5100 over the D7000.]
Take care, Glen