Canon or Nikon. Olympus is OK, but the digital media cards (xD) are almost twice as expensive as the CF cards you'd use with a Canon. Kodak's are more user-friendly, but the image quality is not near the quality of the Canon or Nikon because of cheaper lenses. I haven't looked into Sony at all so I can't commend either way.
I got my first camera, a Canon Powershot S1-IS, last winter and LOVE it. I talked to my boss for months and together we looked at many options before I decided which one I wanted. He has 35 years of photography experience and only recommends Canon or Nikon. Those two brands use the best quality parts. The S1-IS was $400 and then I got an $80 three year warranty, so with tax it was about $500.
Basically there's a few major factors in each digital camera and you have to decide what combination is right for you. First, the megapixel. My camera is a 3.2 which prints up to 8x11. This is plenty for me b/c I don't plan on printing huge pictures. Also, the greater the megapixel, the more memory card space you need, the faster memory cards you need, and the more space on your computer you need. If you're not planning on doing anything semi-professional, you don't need more than 5 megapixels at most, and 3.2 is fine.
Then there's zoom. Ignore digital zoom, it is evil, don't ever use it. The optical zoom is what's important. My camera has a 10x optical zoom, which I think is as good as you can get with a point-and-shoot. It also has an image stabilizer, b/c the father in you zoom, the harder it is to avoid taking blurry pics. I got this b/c I took the camera on safari in Africa and also to all my U2 concerts. So it really depends on your needs.
You don't really have to worry about "speed" with a digital camera. You can use various auto settings, or manually set everything yourself. You can also use a sports setting which allows you to take like 12 pictures continuously. You can also change how long the camera displays the picture after it's taken, so you don't have to wait before taking another.
