New Fan Switch Question
Yes, run a short length of new white wire to the white terminal of the switch and add it to the wirenut (grommet) of the two white wires.

One of the black wires goes to the black terminal on the switch, the other black wire goes to the red terminal on the switch.

It won't damage the switch if you mix up the red and black wires, it simply won't work and you'll just have to swap them around. 

You could restore power and use a non-contact voltage detector or a multimeter to find the hot (black terminal) and switched hot (red terminal), but that's just more work requiring tools you make not have. 

Mark the switched hot (red) wire with some electrical tape to help remember which is which in the future. Red electrical tape would be perfect, but any color is better than nothing!