I have a Fujifilm FinePx WP Z and love it. It was about $150, takes great pictures, has 18x zoom, takes movies, and goes under water. Really. I've had it in the lake, the pool, and the ocean. It is great. We take it on bike rides and just about everywhere we go. Fits nicely in my pocket or purse.
It has lots of settings that will affect the shutter speed (type of flash, indoor, outdoor, sports, nature ...)