The Dipsea Trail is easily the best hike we've done in the Bay Area and the trailhead is literally across the street from the wedding venue. It has everything: forests, waterfalls, ocean views, etc. There are trails for all skill levels, but the "Dipsea to Steep Ravine" trail is the best. Note that it is 7 miles roundtrip and rated "moderate", so plan accordingly.
If you're looking to see groves of redwoods, Muir Woods is the place to go in the Bay Area. It is about 40 minutes north of San Francisco and 20 minutes from the wedding venue. It does get busy, and you'll need to reserve a parking pass in advance, but it is gorgeous and we highly recommend checking it out. You can do the main trail loop which takes less than an hour and is easy to maneuver, or you can spend an entire day here exploring the expansive trails. Note that it is handicap and stroller accessible on some trails!
This is where we got engaged! Point Reyes National Seashore is just north of Stinson Beach. If you had an entire month you couldn't see everything it has to offer, it has countless hiking trails and endless natural beauty.
This is our go-to local Italian spot in our neighborhood of Nob Hill. They have great food and even better service. They do not take reservations, but wait times are usually around 20-40 minutes. There is a $20 corkage fee, so check out the convenient store next door and grab a bottle or two of wine if you want some 'off menu' libations.
If you're looking for remnants of hippie counterculture, Haight-Ashbury is the place to go. Beautiful victorian homes, cool shops, and plenty of peace and love. Amoeba Music is a classic record shop, Sparrow for lunch (check out the back patio), and Golden Gate Park is just down the road.