Well, of course rechargeable batteries for cameras exist! Where would we be without 'em?
I'll ask a few things: family and vacation; do you have little children that are always moving around? Do you plan on doing any action photography? Also, what is your estimated budget?
Will try and explain any questions you have about lenses, focal range, etc.
PS. Digital zoom is mostly and more or less a marketing gimmick - disregard it's existance. Optical zoom is what you really need , when the lense moves back and forth, as opposed to digital zoom, what it really does is enlarge a part of the image, thus a kind of simulation of what optical zoom would do.
The camera looks at the image, then digital zoom crops out wherever you zoomed in, and blows it up size-wise. This can also be done in Photoshop, etc. When you do this, you loose quality. While it might look mighty fine on that tiny LCD, it won't on the screen.