Funny thing...I get emails every now and then, from friends or FB acquaintances that want to know, what's a good (non-professional) camera to buy? And I always give them the same answer...a Canon Rebel. Since I'm a canon girl myself, I think this is a good starting point.
You can buy a Rebel Kit for around $650. Also, don't be afraid to buy used from a power seller on Ebay OR I buy my equipment from Adorama. You can buy new or used equipment and with their ratings system you know what kind of shape it's in. I completely trust the! I just bought my brand spankin' new camera from them and I could not be any happier!! So go check out Adorama!!!
A Rebel is a DSLR camera with some great features. It may not have all the bells and whistle a professional grade camera has, but It's a VERY nice camera for a non-professional.
Now, I must say...once you start using this camera, I'm sure you'll be amazed by it's quickness and sharpness and other capabilites, but if you don't understand the basics of photography, you're still gonna get the same type pictures as your little point and shoot. Sure your images may be a little more vibrant and sharp, but you still need to learn your camera. So once you get it, do me a favor and take out the manual (you know, that little white book that comes inside the box) and READ it!! Heck, watch some videos on you tube. I think you'll be VERY surprised on what you can learn! :)