To start with, there are many brands and models of Digital cameras on the market and if you are just starting out in photography, but hoping to get serious, it is entirely up to you, but I would strongly advise considering a SLR type camera. Some of the newer digital "point and shoots" have much to offer, I know, but sometimes just for the sake of a few extra dollars you can have a camera that will not only allow for full or semi-manual operation and the ability to interchange lenses, but it will also allow you to have more control over the photos you produce.
When you go out to purchase your camera, make sure it is from a reputable photo equipment dealer and not just an electrical store that sells cameras on the side. You might get it for a few dollars less, but you'll find the after sales service leaves a lot to be desired.
Let the sales assistant know that you are just starting out and that you are not necessarirly looking to buy the top of the range SLR, but something of quality, but more suited to a novice photographer. Get them to give you a brief run-down on the benefits of the camera they suggest and at current rates, you should not have to pay more than $750 to $1000 for it.
If you have decided on a SLR type camera, make sure you also purchase a UV filter and fit it onto your camera lens, for no other reason than for the protection of your lens. This will eliminate any scratches, dust, salt spray or finger marks from contaminating the front element of your lens. Keep it on at all times. It is far cheaper to replace a broken filter than a new lens. Remove it only when you want to fit other filters.
Generally, the lens fitted to cameras these days are of a zoom type and could be in the range anywhere from 17mm to 200mm. This is a reasonable range to play with. It is not a "fast" lens, but it will still allow for many photographic situations and will be quiute okay for anyone just getting a feel for this great pastime.
What's a fast lens? See my chapter on "Lenses"