H.W.
Having been in the military (we called the companies of new recruits with shaved head 'swinging d*cks' so be warned) ... if you must do it,do it. If not, consider a very short buzz cut.
Do you have the skull shape for a good-looking shaved head? That's one reason *I've* never done it. Seriously... If I were you, go to a younger barber who has a good reputation and have them give you a quarter-inch buzz cut. THEN, when that's done, ask'em directly-- "do you think with the shape of my head that shaving it would look awesome, or not so much?" If you are going to look great with it, then deal with the eyebrows afterward.
And do use tons of sunblock-- or wear a hat. I don't know how 'cool' that will look, but as the daughter of a balding man, the Red Head look is also going to get you comments.