I'll let the experts deal more with the specific questions, I just want to add a couple of things from another beginners point of view.
Without trying to sound mean about it, I think the way your asked your question might be setting yourself up for a bit of disappointment. I've been using my DSLR for about a year now, and I've been trying to learn as much as possible about it, but the fact is that I am no where near to taking shots like some of the people here on this board (and it's very possible I never will be). Many of the people here are experts who have dedicated years and years to learning and perfecting their skills. If I start using them as a comparison to myself, I find that I can get quite disappointed.
I say that simply to point out, that instead of trying to take pictures like many of the people here, make it a goal to start taking better pictures than perhaps you are now, and to advance from there. Thankfully I can say that in the past year, I have indeed seen quite a bit of improvement in both my understanding of, and actual taking of pictures.
There is still so much more to learn, and so much to work toward, but instead of continually comparing myself to the experts (which can be discouraging), I am comparing myself to where I was a year ago (which is actually quite encouraging).
I hope that doesn't sound mean or rude, but I just want to help you to avoid possibly setting yourself up for disappointment if you don't see the results that you might want to see after some time (basing it on how the experts do).
As for the best way to improve, I would suggest once you decide on a camera, slowly go through the instruction manual, and work on each and every aspect of the camera. Even if it's something that you already feel comfortable with, work your way through the manual. After that there are a ton of great books that expand on each of these concepts (the one that gets the most recommendation here is Understanding Exposure by Brian Peterson) These books can help you advance leaps and bounds in your shooting. Also, be sure to use the camera until your family gets sick of the site of it
Good luck in whatever camera you finally decide on, and enjoy your journey of learning how to best use it. Even though I'm still a beginner, I love taking pictures, and look forward to every chance I get to practice and learn something new - I know you will too.