Head over to the other side of the red bridge turn left just after it and follow the road and some point you can stop and take a photo of the city framed by the red bridge. Haight-Ashbury is great from people shooting and the painted ladies row of houses.
The hills across the Golden Gate Bridge are called the Marin Headlands. To save you from going wrong and experiencing a maddening event, you don't actually turn left after crossing the bridge. You take a right hand slip road after the car park to a viewing area [which is worth stopping at first]. At the end of the slip road turn left and take the tunnel under the main road then turn left. Watch carefully for the road on the right up in to the hills. It's easy to miss and you'll find yourself going back across the bridge [toll to pay in to the city]. Take a look at Google satellite pictures and study it. Print pictures if necessary.
In the opposite direction towards the Pacific is the high city viewpoint called Twin Peaks. It may interest you.
Think about a city tour, as in one of these
http://www.allsanfranciscotours.com/city-tours/sf-city-tours/sbc-2637
The open vehicles are good for photography. I did one which is no longer listed on this site but there's a similar one. I also did a DUKW tour - amphibious landing craft which drives in to the bay. Both were well worth it and you'll see areas which you would otherwise miss. They'll depart from the
Fisherman's Wharf area, and pre-booking isn't essential.
When I was last there, 4 years ago, the painted ladies houses were a disappointment. Sadly in need of some paint.
Coit tower is worth a look but be warned car parking is a problem and may take some time. I could say take a taxi there which is a good idea, until it comes to leaving and the area you are in isn't flush with passing taxis. I'd also suggest going late afternoon and staying until dark for some good sunset/night time shots. Which makes the taxi back even more of a problem.
If you want to go on one of the trolleys then skip the massive tourist queues at each end [Fisherman's Wharf and Union Square] and get on part way along. Or try one of the non-tourist lines like California.
If you're wanting to unload some money then think about one of the flights over the city. I did a seaplane one and it was outstanding. The website above for this tour says 2 person minimum booking. I went in to a tourist office on Pier 39 [Fisherman's Wharf, upper level ] and they booked me a one person flight with this same company no problem.
Tripod in Alcatraz - I managed without one.