I ended up getting a Fuji FinePix S8000fd. I was close to biting the bullet and getting a digital SLR as I love taking photos but (a) they're really expensive, (b) my partner wouldn't know how to use it and wouldn't have any desire to learn, and (c) I didn't want to be carrying lenses around with me.
The Fuji provided an 18x optical zoom and then a further digital zoom, with various settings. It seemed like a decent bridge between our existing camera and a Digital SLR.
The action shots on the Fuji is just a case of switching to "Sports" mode, though you can, if you want to, change the aperture, shutter speed, ISO, white balance, and more besides.
It's a decent camera.