I recently visited the golden gate bridge and stopped at both north and south sides. One of the main stops in the north side is the vista point which you find it immediately to the right after you cross the bridge from the south. You get a good view of the bridge from there. You'll also find the statue of the lonely traveler and a distant view of the Alcatraz prison island. The parking lot is somewhat small and you might have to wait a little bit to find a parking spot. There is also a side walk along the bridge.
From the side, you can drive/hike along the beach. We drove along Marina Blvd and Manson St (North East) and Lincoln Blvd (North West). If you want to take photos, it is better you go there in the morning or late in the afternoon close to sunset. In the afternoon, the sun directly falls on the bridge and it is hard to take any good photos.