V lived in North Beach with Stephan for a few years (on and off apparently), so you can definitely check that neighborhood out. She's also been seen running through the Palace of Fine Arts in the Marina/Presidio in her "Pleased To Meet You" vid. You've also got the beach that V ran through in the vid, called Baker's Beach, which looks cleaner than Ocean Beach to me, and the famous GG bridge.
My mom knew a few co-workers that loved a place called "The Stinking Rose", which is a garlic-themed restaurant. I usually head to North Beach for Italian or this Asian fusion restaurant (finer dining) called "House". If you're celebrating your anniversary, you can consider adding it to your restaurant list. Food = delicious, but service can be a little snooty.
But yeah, North Beach practically intertwined with Chinatown, so you can always roam there, too, for baked goods, cheap souvenirs, and generally cheap, tasty Chinese.
If you find yourself looking for great breakfast, one of my favs is "The Village Grill", within Outer Sunset/ West Portal. It's fairly crowded around 12-1pm on weekends, so be sure to get there early if you don't like waiting 15 minutes or so. "Squat & Gobble" is another option a few blocks away. If I had to rank my Top 3 neighborhoods, they would be:
1) Sunset (especially West Portal and Irving!)
2) North Beach (a number of yummy eateries, like "Golden Boy Pizza")
3) Richmond (Clement St, etc)
And as always, try avoiding touristy areas, like Fisherman's Wharf, Downtown, Embarcadero, and the Zoo, unless you don't mind spending extra $$. Best for you to do your research before heading out there.
AND,.. if you're interested in overlooking SF, try Twin Peaks. Arguably the best view of the entire city, although I've never had a problem walking by Portola Ave either.
My last word is on museums. My favorite has always been SFMOMA, although the ticket prices are a little steep. I enjoyed the Frida Kahlo exhibit they had 2 years ago or so.
But anyways, you can always search on Yelp for other dining suggestions, popular shops, and other attractions. Have fun!!