I want something that can handle indoor shots when the kids are far away and sports action....but if taking it out of the box, putting it on auto, and pushing the button isn't going to get me good pics, it won't be worth it. I don't want to think about lenses, extra flashes, ISO, white balance, or any of that other stuff.
I'm afraid that the camera you are looking for does not exist.
Taking indoor photos and kids' sports photos is not a zero-effort proposition. If you aren't interested in learning photography you won't get good photographs under those conditions, whatever equipment you buy. Unfortunately even if you are willing to spend the months or years necessary to learn you still won't be able to get fantastic results without some expensive equipment. What you want to do faces the same challenges as wedding and sports photographers. These people spend years learning their craft and spend a lot and a fortune respectively on their equipment.
Which is not to say that it is not possible to get a good P&S camera which will give better results than you are getting at the moment.
I think your best option at the moment is probably the Canon G11 or Sony H-20, also purchase at the same time some lowered expectations and you'll be good to go. :-)