I don't know home prices, since my family rents, but $750 is modest for a SF home. Sounds more like a condo budget to me.
These are some general pointers - if you want something more specific, let us know what you want in a neighborhood. E.g. why do you want a second home in San Francisco?
If you need mass transit to the Peninsula/South Bay, you need to live near the Caltrain station (near the ballpark) in SoMa. There are plenty of condos in that area, though I don't know if the neighborhood is any good to live in. (Not a lot of green space, for instance.)
For the East Bay (and down to SFO, but not much further), you can live anywhere along the BART line - from the Mission to SoMa to the financial district.
Potrero Hill isn't too far from either train line, and seems like a good neighborhood - close to the Mission, close to SoMa, etc. It's also supposed to have some of the best weather in San Francisco.
Mission Dolores is also a great neighborhood, and close to BART, busses, etc. (Actually, most of SF is accessible to public transportation like bus, streetcar, etc.)
I live in NoPa (western Western Addition to purists) and am a big fan of that neighborhood. It's about as central as you can get - 15 minute drive/bike from just about anywhere in town, and walkable to Pac Heights, Japantown, Golden Gate Park, Alamo Square, Haight, Lower Haight, Castro, etc.
Richmond and Sunset are safe and probably have places in your budget, but they're remote. They're also foggier and colder than the eastern half of the city.
Most of SF is safe. Just stay away from gangs and (non-medicinal) drugs, and you'll be fine. The Tenderloin is the "bad" part of town, and parts of Western Addition and Mission have bad reputations too, but they aren't really that bad. They get their share of property crime, but not a lot of violent crime.