T.S.
I can't speak for those particular neighborhoods (though I know the Mission is funky and fun) but SF is a small city so it's easy to explore the whole area.
With the kids-
walk around pier 39/fisherman's wharf (touristy but fun) and Chinatown, hit the playground/carousel at Golden Gate park, the SF Zoo, the Exploratorium (sp?) and the Ferry Building. If the weather is nice walk or bike across the Golden Gate bridge or check out the beach below the Cliff House, very cool. And ride a cable car of course!
Without the kids-
Tour Alcatraz, shop in Union Square, have dinner in North Beach, go bar hopping in the Marina and go see Beach Blanket Babylon, it is the ultimate San Francisco show experience! If you like art I LOVE the museum at the Legion of Honor at Land's End, a beautiful museum in a spectacular setting.
Have fun!!!