How do I stop lights from flashing when I turn on my table saw?
When an electric motor turns on, there is a thing called "inductive kick". What this means is that for a second at start up, the motor looks like a short circuit and draws more than the rated amperage. Voltage is dropped as well. This is corrected by huge capacitors in an industrial situation. In your case, the combination of large current drain and a resulting voltage drop causes the ballast for the florescent lights to fail to deliver proper voltage to the bulbs.  This is common.  I would move the plug for the lights to another circuit. The only other thing that will cure the problem is a motor starting capacitor.