I'd recommend an older second hand DLSR (you'll get a better deal and might find one that comes with additional lenses). Generally speaking a better lens on a weaker camera will give you better results than a cheap lens on a good body. Nikon vs Canon is subjective, buy what you can afford and what feels good in your hands. Don't disregard other makes such as Sony either. Knowing how to use a camera will get you better results than just buying the best and hoping it will do the work for you Any entry level DSLR with a stock lens will be good enough to get you started though.
You could also try picking up a 35mm film camera and learning the old way. You can get really good film cameras with decent prime lenses (a fixed focal range, meaning you can't zoom in/out) for a very low price nowadays. You'll need some way to develop the film though, but it's a great way to learn photography as it forces you to think about what you're shooting, how much light there is, etc.
I often take an Olympus Trip 35 out with me. Essentially it's a point and shoot, so most of the technical stuff is handled by the camera, but I can manually set the aperture to adjust the amount of light hitting the film. which lets me focus on the subject matter of what I'm shooting rather than worrying about the setting being correct. It lets me enjoy the actual act of photography and generally gives me decent results. The beauty of film photography is you never quite know how your shots will develop, it's scary and exciting at the same time.
Oh man.... the brand of the camera isn't really that important these days (Nikon use Sony sensors for example), it's the lenses which make up most of the difference in terms of picture quality. Having said that you do nee to understand 'how to compose' a picture and imo it's a hell of a lot easier to learn how to get a good picture than when I started out... there was no dslr so it was all 35mm film so you couldn't afford to waste pictures and my first camera was totally manual with focus, aperture and shutter speed all having to be set up manually.
@Wardy You can include Sony in there as well, you can actually get a better Sony camera for the same price as Nikon/Canon and they're pretty good cameras too because they're originally based off Minolta and still take all their lenses.
I've got an old cosina c1 (my first slr, fully manual, based off pentax), minolta and nikon which are all film based.... still haven't got round to a dslr but keep looking at sony alpha 58 or 68 because they use my lenses where as nikon requires higher end models to make us of my lenses (no motors) so is quite expensive. Mind you I might just say screw it and buy a canon and replace the lenses as it's likely that won't be any more expensive lol
To the OP - what sort of budget and what sort of photography are you interested in doing etc all makes a difference when we make suggestions so the more info you can give the better