Walking Across the Bridge
The best way to see the Golden Gate Bridge is walking along the pedestrian walkway (or biking along the other side if you prefer). The pedestrian walkway is on the east side of the bridge (on the bay side) and is open only during daylight (the hours are extended a bit in summer). The path is level and dogs are allowed if on a leash.
The entire span of the bridge is 1.7 miles between viewpoints. If you do not feel like walking the entire bridge, walk as far as the first support where you can look straight up to see how massive the bridge is.
If you prefer to ride your bike across, you can do so 24-hours a day, seven days a week (although on weekends bikes are only permitted on the west side of the bridge due to the larger pedestrian crowds on weekends).