You can get one from any major brand for that price to do what you want. The trick is to turn off the auto focus mode. So you would have to set the shutter speed, aperture size, etc. yourself. Also, get a high speed memory card for the camera, it makes a big difference.
I personally prefer canon or panasonic for their lenses. i have a panasonic lumix DMC-ZS#, i can't remember the number. i got it a few years ago for around that price. it's small enough to fit in my purse, i can set it to autfocus or i can set the settings myself and save these modes for different situations. and of course the video mode is also great.
what slows down the camera is the software that makes sure you get a good picture with every click. it tries to figure out the settings for you depending on your environment.