I'm a total amateur hobbyist who's looking to become a better hobbyist with time. My next camera will definitely be a dSLR, probably a Canon in the affordable range.
You're right about the better quality cameras. But keep in mind, Kodak makes a vast line of cameras. Their low end ones don't even resemble their midrange or high end ones. I myself don't want a Kodak, as they don't make consumer-level SLRs, but I bought a Kodak 4.0 megapixel for my wife a couple of Christmases ago. It takes good pictures over and over, if you are mainly taking fairly family pictures and such.
From what you say, though, about not liking to use flash, it sounds like a high-end "prosumer" or a dSLR is your cup of tea, because you can crank them to 800 or 1600 ISO, require less light, no flash, and still get clear pictures. You just cannot, cannot, cannot do this with any low end camera. Even my DiMage A200 8.0megapixel takes very grainy pictures at ISO 800. I would use 400 ISO only for pics I intend to make into 4x6 prints (max) or publish on the web. Whereas, with a Canon Rebel XT or XTi, you can take ISO 1600 shots, no flash, and get clear low-noise pictures that are suitable for 8x10's if you want....and never say you'd never want one... you just don't know what picture is going to turn out great until you look at them later
If I had known about the need for high ISO capability, I would have probably held out for a dSLR from the beginning, but I really didn't understand the NEED for this until after I shot a couple of thousand pics. I kept trying to stay at ISO 200 or lower for clarity and kept getting blurry shots due to slow shutter speeds.... Once I figured out that cranking the ISO fixed this, I was dismayed at the digital noise. The only way to fix that is to get a dSLR. You can get a Canon Rebel XTi setup with a basic lense for under $1,000 and even if you leave it on "auto" you will take better pictures than most people have EVER taken.
Barring that option, I'd look at the high-end of the regular digitals (non-SLR) if you are sure you want to rarely use the flash. The latest one from Sony is great, but again, expect to spend $600-800.
It's well worth it, though. Once you get just one nice series of gorgeous pictures, you'll happily take every point-and-shoot piece of pocket junk and throw it right in the trash! Imagine how great all your old pics would have been with the right camera
Now, that all said, here's a couple of examples of what you can do with a high-end non-dSLR, e.g. the DiMage A200 (discontinued) or perhaps one of the current Nikon, Canon or Sony's. FULL 8mp images.
HIPPO AT ST. LOUIS ZOO IN 2005
GLASS KISSES AT TEXAS RENAISSANCE FEST THIS WEEKEND
Caveat: the image of glasses was taken at ISO 400 w/ no flash, but I had to use an image noise reducer (Neat Image freeware) to get the "glass" smooth image you see here... But, if you want to avoid an SLR, there's your ticket
Good luck,
Michael