Write a sketch that reads the pushbutton and detects when the button changes from 'not pressed' to 'pressed'.
Write a sketch that flashes an LED, based on the 'blink without delay' example.
Provide a suitable power supply for your servo, remember to connect the earths and use the servo library to sweep the servo through the range of positions you want at the speed you want.
Once you have all the elements, you'll use the button code to trigger the start of each movement sequence, flash the LED while the movement is in progress and stop flashing when the movement is complete.