Although you'd use Doppler shifting from the laser signal to measure the speed, you're going to have an extraordinarily difficult time doing it.
Let's suppose that you're ABSOLUTELY SURE that the ball is going to go through a specific place (good luck with that on the green) like a gutter, or if the golfer is putting the ball into the hole and he/she has a precise aim that'll go right through your setup. That's the simplest case; you'll only need a laser and a photosensor. When the ball cuts the beam shining on the photosensor, it registers that the light isn't coming in any more and that produces a signal. Then you only have to wait a few milliseconds until the photosensor registers it again. The signal from the photosensor goes into an ADC connected to a recording device (for example a serial port on a PC). You'll need software to convert the serial input and process it appropriately.
But if you're trying to catch a ball flying and measure its speed with a laser beam -- then you can foggedaboudit.
Firstly, the objects which you want to measure are round; they don't have sharp edges reflecting the light at a neat angle. That means that the light shining on it will reflect into whatever direction it feels like it. Good luck getting the return beam right into the sensor (unless you're superman and can fly around the ball fast enough to put the sensor right in front of the reflected beam). Secondly, you'd need an extremely precise (and expensive) sensor to measure the wavelength of the incoming laser, and it'd have to be sufficiently sensitive to measure the difference in wavelength from the emitted beam to measure the Doppler shift. Given that the speeds are very low (a fraction of the speed of light), it won't change much. If you think that a golf ball goes at, say, 30 meters per second, and that the speed of light is 300 thousand kilometers per second, you're talking about a Doppler shift of 1 part in 10 million -- which is very likely smaller than the wavelength spread of the souvenir store beam. I'd be ABSOLUTELY NUTS to try such setup with a laser to measure the speed of a moving object!
If you REALLY insist on using Doppler shift to measure the speed, then you need a setup that emits a WIDE beam and also that can capture the bouncing signal from ANY position (i.e. not being dependent on the geometry of the object).
You could try a sonar setup, but the ambient noise will suck so much you'll be crying tears of frustration.
If you can, get your hands on one of those police radars and try it. The source and detector are all conveniently and neatly packaged and you just have to point and shoot; that'll be the most work you'll have. Otherwise, you'll be banging your head on the table with frustration at how difficult it'll seem.