Whether or not you should even get a DSLR camera would depend on the type of photography you will use it for.
Is closing on a subject far away important as well as taking a picture of a wide landscape? You should probably go with a Compact zoom such as the likes of the Canon PowerShot.
If you are willing to approach the subject and want a higher quality image then you should be looking at fixed "prime lens" likes of the new Fuji Pix X 100.
If you are still leaning towards a DSLR camera but want the convenience of not having to change lenses then you will want to get it with an "all-around or walk-about" lens, something that can go from wide-angle to a telephoto range. These lenses are optically inferior so you will have to sacrifice image quality for convenience. You will pay premium for a half decent "walk-about" lens and it will never be as good in a given focal length as a prime lens.
Ultimately, if you want a really smooth, crisp image then we should be discussing lenses not cameras.
You could always get a inexpensive DSLR with a mediocre "walk-about 18 - 200+mm" lens but in this case that's just an over-sized "compact zoom" (what you refer to a digital) camera. However, purchase an inexpensive fast 50mm lens and you will be able to achieve stunning artistic shots of your family with great bokeh and all. (Canon has one of those lenses for $100, couple it with a Canon Rebel and the 18 - 135+mm lens and you will not be disappointed)