Someone asked me this on Twitter the other day. (Which amused and flattered me because how should I know? But it turns out I have ideas on the subject. No one fall over in shock.) Her question was, specifically, should she start off with a DSLR? Not having done it both ways because you can't, I'll just offer up my experience and resulting opinions. I didn't even want to be a photographer. I actually was against it as a hobby because I grew up having to crawl under my dad's darkroom table to take a bath (it was built over the bathtub, leaving about 8 inches of space; my family takes geekery in all forms VERY SERIOUSLY). However, digital cameras ARE gadgets, so I eventually bought one (a Nikon Coolpix L11).
To my shock I found I liked what I saw through the viewfinder, and I liked what postprocessing could do for me. I could take what I saw in my mind's eye and make a photo out of it. People started saying I had a good eye (my dad said it was genetic).
Then I got an iPhone and heard about iPhoneography, which appealed to me because I always had my phone and always had a deep affection for street photography and hip shots, so I started and finished an iPhone365 project (I missed only 1 day, I had a really good reason, and you can see the rest here). Most of the year was shot with an iPhone 3GS, so it had some white balance--adjusting capabilites, but otherwise shooting with an iPhone forces your hand: you have to find interesting subjects or interesting ways to see uninteresting subjects, because the camera is not going to help you by turning out a technically brilliant photo to save you.
I then started lusting after a DSLR, but only after I'd been taking pictures regularly enough to see the shots I was missing because of technical limitations. I really don't think I would have seen or appreciated the subtleties of photography if I hadn't spent a year taking photos with a cell phone camera and experiencing intermittent frustration with its lack of technical shine.
I got the DSLR at long last (a Nikon D3000) and spent a long time figuring out how to use it. I'm still figuring out how to use it. Just recently I added a high-end point-and-shoot (a Canon PowerShot S90) because having an iPhone and a DSLR put me in the most frustrating photographic world: being used to portability yet knowing the shots I COULD be getting with my giant DSLR. If I know I'm going to be taking photos, I take my DSLR, but I have this tiny Canon now that has full manual settings, takes some truly impressive photos, and is not much bigger than my phone (so I usually have it with me)---and again, I would NOT be able to use the Canon the way I do had I not gone through the gamut of the other cameras.
Readers are probably hoping I'll come to the point soon, and since I'm feeling PRESSURED, I guess it's that new photographers should start with what they have available. It seems whatever equipment you pick up, you will learn its limitations and strengths and then pick up other equipment and see where it fills in the gaps, all the while developing a firmer sense of what you, the photographer, want your device to capture for you and how. If you've got a good eye, a lesser-quality camera will only make it better. If you don't, a $5000 camera won't help you.
The answer is, then, this: the kind of camera that you take a TON of photos with, all the time. Any camera will probably do.