I've had several Sony Cybershots. I like them. Good optical zoom. Nice sized screen.They use memory sticks, and since I already have memory sticks it works nicely. When my current memory stick is too small to fit the photos I like to take, I just buy the cheap brand (SanDisk) version of the next bigger size and it works out nicely. A 1 GB stick should be plenty, although we use a 2GB and love it. There is no in camera memory for the versions I've used, which I like, but then again maybe in camera memory is faster or something. I like the burst mode - a necessity for taking pictures of squirmy babies - it takes up to 8 shots in succession. The movies in our current camera (W-3 I think) are good quality and easily viewable on our computer. Our previous version the movies were too small to be interesting. However, both the movies and the burst mode tend to be dark unless the lights are on (e.g. taking a video of the baby going to sleep doesn't work well).
So in summary, I am quite pleased with our Sony Cybershot W-series and it appears to fit your requirements. However, I have very little experience with any other brands so I can't tell you how it compares.