I have the Kodak Zx1, and overall I am happy with it for the price (<$250AU back in 2009, cheaper now). It comes with two rechargeable AA batteries (low self discharge AFAIK) and a basic charger.
Its main attraction to me is that it is pocketable, rugged and dustproof/splash resistant. The batteries last for around 2hrs of HD recording.
I don't really have a lot of preference other than it works nicely with a Mac.
Files are in .mov format and import into iMovie with no problems.
The outside front of the Zx1 has a very thin 'rubbery feel' coating which allows good grip and the camera is easy to operate one-handed. Startup time is just over 1 second.
Its biggest problem to me is the lack of Image stabilisation. If you are simply holding the unit still and videoing a moving subject such as you would do with a golf swing, the quality is OK, but as soon as you move (even walking) like following your daughter around, the video becomes shaky, and forget about mounting it in a vehicle (which is one thing I wanted to do). It does have a tripod mount threaded socket (1/4" Whitworth bolt fits perfectly).
Some more recent magazine comparisons rate the JVC Picsio GC-FM1 as being the best of these kind of video cameras, so maybe check that out too.
HTH!