If you're serious about making this a hobby, get an entry-level dSLR body instead of a point & shoot, no question. If you go for last year's models or look around used, you should be able to at least approach your $300 price point (at least, you'll be able to see that point over your shoulder from the point you end up at).
As you say, there's so much complexity that the beauty of photography -- the basic three factors of exposure, and then a single click to capture an instant out of the universe -- is easily lost. It is a shame there's not a digital equivalent of a bare-bones all manual Pentax K1000 film camera in digital. But, I'll make a somewhat unorthodox suggestion that may appeal to you given what you've said:
Get that entry level digital SLR I mention above (I like Pentax, but you can't go wrong with Canon or Nikon, except you may want to rule out the Nikon D40 because of what I'm going to say next).
Then, get the best normal-range prime lens you can afford. That means somewhere between 30 and 50mm, with no zoom. The Nikon D40 can't autofocus with the basic versions of these lenses, so that's why I suggest staying away from it (it's otherwise a great little camera).
Put the camera in M mode, and make yourself leave it there until you feel comfortable with what it's doing. Then, after a little bit if you want you can move to Av mode (aperture priority) to let the camera worry about shutter speed and exposure while you worry about composition.
If you feel ambitious, switch the camera to manual focus, although this tends to be hard on the less-than-stellar viewfinders of entry-level dSLRs.
Then, take your camera out and take lots of pictures. Get to know the field of view of your lens and what it can do. Get to know how to get exposure right. And then work up from there.
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