I could not have said it better.
The only thing I would add is regarding lenses. They are going to drive you over your budget, later if not right away. I would strive for your initial lens to be a modest 18-55mm zoom -- it may even come with the camera in a kit. It will give you a little wide angle and a little zoom for a modest price. (Just about everything else will be more expensive -- $300 and up for the lens alone). Please do not make the mistake of getting an aftermarket lens because it is cheap -- it is cheap for a reason, your pictures will show the difference. I recommend sticking with the Canon or Nikon optics that match your camera choice.
Actually the Nikon D5200 with that same lens is on their website for $799.95. It would be a good starter for you. If your requirements are modest, you may not need any more.
I am much less familiar with Canon, but if you find a combo you like there, I do not think you could go wrong.