New to this board but not new to photography at all... While OP's question is quite old, I'd like to chime in as well for newer readers.
If it interests anyone here, my sig has a link to
my blog for a lot of photography lessons, feel free to visit, it's free.
If you have the funds, get one DSLR and a pocket P&S for various reasons.
A DSLR would certainly open new doors when it comes to photography. While most higher-end P&S cameras have DSLR-like features, the speed, intuitive controls, general image quality (I'm talking about the data, not the photo itself), etc all helps propel your hobby.
Now if you only plan to have a camera to snap day-to-day stuff, your kids playing around, etc without being concerned about photography fundamentals, so to speak, then I'd strongly suggest a high-end P&S like the LX3, Canon G10, and the like as they give you excellent detail and quality while being compact-do-it-all as well.
A DSLR will open your eyes to photography but it'll also empty your wallet quite a bit as it's inevitable that you'd be buying more and more add-ons such as lenses, flash, tripods, etc.
So I suggest you get a high-end P&S first, as even if you do decide to get in to photography seriously, the P&S will still be extremely handy and won't hamper your learning process as the gear is technically more than sufficient for great photographs.