As previous posters have mentioned, there's no need to have a car in either Seattle or San Francisco. Hire it for your coastal drive, return it in SF and fly back to Seattle. If you do end up with a car, SF is a very compact city, so most of the areas we frequently recommend for guests are convenient to public transit -- Union Square, Embarcadero, Fisherman's Wharf, Lombard Street. (If you don't have a car, there is less reason to stay on Lombard Street, though...)
Keep in mind that we Americans can be very intimidated about driving in the UK... "wrong" side of the road, roundabouts, zebra crossings... all of that seems very mysterious and scary. Once you get used to it, however, it's easy! The same will be true here.
The only thing you might do is find something that lists common road rules in the US. I don't know why rental car companies don't give these out. In California, you can generally turn right on a red light if there is no traffic coming from your left. If you come to an intersection with four stop signs (not lights), the car that approached the intersection first continues first; in case of a tie, the car to the right goes first. You can Google for images of stop lights, pedestrian crossings, common road signs, etc. That might also ease your mind a little bit to know what you're going to face.