On the size and weight issue -- It comes down to whether you intend to carry the camera for long periods, whether it's a strong preference that it can fit in a pocket versus small bag versus around the neck/large bag. And as to weight particularly, a bigger camera lens combo can wear some people down if carried for an extended period. On the other hand, if you are mostly using the camera around the home, size and weight are much smaller issues. But if you're walking 12 hours per day with your whole camera kit, size can be a huge issue.
For up to $1000, you have lots of options.
The high end compacts -- like the Rx100, are outstanding. Image quality very very close to a typical dSLR (in some cases better), yet it fits in your pocket.
Then there are "mirrorless " cameras -- the Nikon 1 is small with amazing autofocus speed ( compared to other non-dslrs. The micro4/3rds cameras have lots of great lens options. High image quality, all in a much lighter and smaller form. (Not quite pocket, but much smaller than dSLR ). The Sony NEX system uses the same sensor type/size as most dslrs -- so outstanding image quality, yet small and light.
In the dSLR world-- you can get entry level or even mid level, and perhaps a couple upgraded lenses.
Remember... DSLR and mirrorless cameras -- to get the most out of them, you end up adding lenses over time. Many people do stick with just the kit lens, but then the camera is not living up to its potential.
In terms of which dSLR -- Canon, Nikon, Pentax, Sony ( which is dslt technically) all make excellent cameras with slightly different strengths and weaknesses. But it's hard to g wrong.
First figure out which type of camera you want, based on ergonomics, shooting style, etc.
Then worry about picking which model.