My first caveat for first time visitors even if it has nothing to do with what they asked- bring warm clothes. Seriously.
Golden Gate Park in in San Francisco. It's similar to NYC's Central Park in that it's a huge amount of green space reserved in the city and has a lot of different things to do and see. We love going here, and after 8 years, we still haven't seen everything. You won't see the bridge from here unless you go to the top of the De Young Museum (which is a free elevator ride, although the museum itself is not free) and can happen to see the Bridge. I can't recall if you can spot it from there, but it wouldn't be a fantastic view if you can.
The Marin headlands are on the opposite side of the Golden Gate Bridge from San Francisco. The gorgeous shots you will see of the bridge with the city in the background are often from here. Unlike GGP, there isn't a lot of different things to do here. It's more driving, looking at amazing views, and some hiking.
They are very different places, so it depends on what you are looking for. Honestly, if you have time, I'd do both as they offer different things. If you don't want both, can you explain what you are looking for?
I believe what a PP was referring to with the park ending at the Fort is Crissy Field at the Presidio.