Because I'm actually training for some masters sprinting events I'll have a lot more to say about this in the future, including a program you can follow.
Most serious track, cycle and swimming sprinters do some sort of weight training these days although it may not work for everyone. Carl Lewis, the great American sprinter and long jumper, says that he performed poorly in one year that he did weight training. But then, when you are as naturally endowed as he was, perhaps you don't fool around with it.
Being "fast" is pretty much an inherited thing and although there are training techniques that can improve speed to some extent, true sprinters are born and not made. Various forms of weight training may help by increasing leg strength and explosive strength and elasticity. Even so, you won't get faster by just doing leg strength exercises. You need to improve your plyometric or "bounce" strength. Mostly you will train for this on the track or hills. Explosive exercises in the gym may also complement this. Box jumps and marches are examples of exercises you could use.
Squats and deadlifts will enhance leg, back, butt and abdominal strength. "Power" exercises like hang cleans and push presses should provide all-round explosive strength. You need to use these exercises intelligently because they are very taxing and best used in a preparatory phase and not too close to competition time.