I am looking for a new digital camera that doesn't break the bank. I am certainly not a professional but I like my photos to look professional.
You need to be specific at what you want to do with it. Just like I can't recommend you which car to buy without knowing anything about you, the same goes for cameras. We don't even know what 'break the bank' means for you, it is very relative.
Definitely if you are looking for high quality, people will tell you to get a DSLR or interchangeable lens camera. The latter is smaller but also slower, if you don't shoot things that move, you won't care about that.
My advice is to read a buying guide that explains to you what the important points to consider are.
Don't get fooled into thinking that you need a professional camera to make fantastic images. Take a look at this pro who takes a $150 digital camera when he goes on vacations.
Sometimes gear is not to be blamed for the photographs to come out "unprofessional".. Before transitioning to an SLR three years ago, I had a really modest kodak camera and used to look at flickr's "interesting" pictures from that camera to get the needed result (by looking at the exif to check the modes they were using and the composition).
Since you probably don't want to get neck deep into photography, nor want to lug a heavy camera around, I'd say grab a great well reviewed P&S camera like the G11 and use flickr to learn taking great pictures with it (http://www.flickr.com/cameras/canon/powershot_g11/)
The question is quite vague, what does professional mean to you? What's your budget? Breaking the bank varies by person. The Pentax 645D at $10,000 breaks the bank for me, but the Canon 5D mark II at $2500 does not, but for someone else, $2500 is way off the line.
In any case, the best way to approach the purchase is based on your needs and desires. To that end, DP Review has a nice tool for plugging in what you're interested in and then recommending some options as a result. That's a place to start, but then you need to look at the potential options online, especially images. At which point, the Flickr Camera Finder becomes handy for seeing the art of the possible.
However, despite all of that, the camera is just a tool. Your pictures will look as professional as you make them, end of story. For some people, you could put the lowest specified camera into their hands and they will produce incredible photos that would make some of us drool. For other people, you could put a $42,000 Phase One camera in front of them and they'll produce a piece of junk. It's not the gear, it's the photographer. If you still don't believe that, consider this: some of the greatest photographs in the last 100 years were taken by photographers using gear that we would currently consider to be utterly archaic.
Net effect, you need to figure out exactly what you want and then use that information to make a decision. Then, if you want professional looking results, you need to practice, practice, and then practice some more. Unless you do both, the camera you buy will not answer your question, regardless of what you choose.