Is the laptop's fan supposed to start and stop very frequently?

The fan is controlled by a thermal sensor.
When the upper edge is reached, the fan starts. It is stopped when the lower threshold is reached by the cooling action.

So, frequent switching of the fan implies you are in a hot environment
(or your laptop vents are not properly ventilated (maybe you are using it on your bed).
The fan will trip on the higher temperatures,
start forcing the temperature down and,
when it sees sufficiently lower temperatures,
will turn off. But, the conditions may push the temperatures high again,
tripping it back into action.

This can also happen frequently when you are doing processor-intensive work.
If your mobile processor supports working at lower power/speed,
the processor intensive activity will push it into higher activity and cause more heating -- which could also trip the fans more often.

Laptop fans are designed to be started and stopped based on sensor data.
I do not think they will get more wear due to that (mine has been working for over 6 years now).
On the contrary, i suspect that a continuous drive might not be very good for the fans life (tho, I am sure there are arguments the other way too).

I think you should not worry too much about the fan.
Meanwhile, you could do something about ventilation and local temperatures to trigger it less often.

Along Jeff's notes, you could also use the SpeedFan app from Almico.