To call San Francisco "dog friendly" is about as controversial as calling it foggy, hilly, caffeinated or technological, etc - it's not unanimous (see: dog politics), but it's de facto :)
Dogs are basically everywhere, and especially Dolores Park, Crissy Field, Golden Gate Park (especially on no-traffic Sundays), Ocean Beach and Fort Funston.
Dogs can ride on Muni with simply a leash (BARTing requires a crate). They must pay and are not officially permitted during rush hour, unless they are service animals.