This is probably the reason why I like staying at Fisherman’s Wharf. They have bike rentals here and you can rent a bike and ride it through Presidio in the Golden Gate Bike Path to the Golden Gate Bridge. This would probably take you most of the day. I would also get off the beaten path and make a short detour to the Palace of Performing Arts.
If riding isn’t for you, take a drive on the Golden Gate Bridge and drive up to Battery Spencer. This is the best view, I think, from above of the Golden Gate Bridge. Please note, that when you drive back on the bridge to SF, you need a FasTrak. If you don’t have one, they usually send for payment through the mail or if you’re renting a car, the rental company will probably charge you. Or you can be proactive and pay online before they send you a bill! I’m sure there’s a Muni that goes over the bridge but not sure if it’s goes to Battery Spencer. From here, I would go through Presidio and then to the Palace of the Fine Arts.
If you have time, I really recommend the Blue and Gold Fleet Cruise. The cruise will take you under the Golden Gate Bridge and around Alcatraz (it doesn’t stop there). Check out the times on their website (they also have discounted tickets on other websites like Gold Star). Or even do a tour of Alcatraz.
If you’re not interested in going to Alcatraz or cruising then I would recommend going to Golden Gate Park. Check out the Japanese Tea Garden or the museums in the park. Drive down a little more and check out the Painted Ladies at Alamo Square. Drive down a little more, and check out the City Hall.
End of day 2.