That helps a lot, thanks.
So, first thing to know is that the quality of your lenses is far more important than your camera body. You will always be better off going to the next level down on the camera body and buying the next level up of lens.
Fortunately, these days there are some really decent zoom lenses that will cover your needs (based on what you wrote).
For shooting cars and such, you're going to want a wide angle lens. For the kids, you're going to want something that's going to let you get a bit tighter in to them as well. So, at that budget amount, I am going to STRONGLY advise against getting a dSLR. It's a system that's on its way out and the few advantages it has left are not needed by you.
As a parent, I can also tell you that you will definitely want the ability to shoot video. Going back now and looking at some of the video I've shot of my kids is priceless. And creating hybrid output (combining stills and video) is really easy with platforms like ProShow Web as well as the software that will come with your camera.
Instead, take a serious look at a mirrorless camera. They're smaller, easier to use, quality is the same as the dSLR bodies, and less expensive. The two I would point you toward are the Sony Alpha a6000 and the Panasonic LUmix GX7. Both have kits that are around the $700-$800 mark at B&H (bhphotovideo.com - no affiliation!). Eventually you will want to add or replace the lens that comes with them but that's down the road. The kit lens is pretty decent so you shouldn't feel like you need to replace it immediately. However ,as far as kit lenses go, I'd give a slight nod to the Sony.
Both the a600 and the GX7 get rave reviews and I think you'd be happy (and satisfied) with either.
Now, in the interest of full disclosure, I am a mirrorless evangelist and currently part of the talent team at HybridPhoto.pro. But I am not paid to promote any brands (in fact, I can't).
