I have a Nikon Coolpix and I also hate how long it takes to recharge the flash. It also takes blurry pictures inside no matter what you do.
I am also researching new cameras with my camera savvy brother's help. He told me to get a Canon SDXXX. (The models change monthly I can't keep up.) The Canon SD's are the skinny almost credit card sized ones that can fit in your pocket and have a 3" LCD screen. They take decent pictures and have a 4x zoom. The camera that I *really* want is the Canon S3, which is a big camera almost as big as a professional SLR. Dell has them usually for about $330 and places like Target & Best Buy have them for $399. It has TONS of options and a 12x zoom. It has a BIG lens and a pop up flash that is supposed to reduce red eye. But you have to be willing to learn how to use it.
A couple things to think about are:
Get a 6+ Mpix. I think 12 is equal to 35mm but unless you want pictures bigger than 8x10 you'll be fine with anything 6 or above as long as you get a good camera.
Most small cameras have a 4x optical zoom and a 12x digital zoom. Digital zoom does nothing more than crop the photo and you loose quality. So never compare digital zoom, always compare optical zoom.
If you just want point and shoot, get any Canon 6+ Mpix in the SD or A series. Nikon and Fuji are also good alternatives. Buy a camera made by a "camera" company. Not Sony or HP. You'll be fine
If you really want to do the research try this website:
http://www.dpreview.com/