Nope, I'm a SF native, and love almost all the neighborhoods of the city. I like different things from travel perhaps than some of the other posters. To me, it's contact with local life on the streets and buses and coffeeshops that's most interesting. Chinatown, Union Square, the Embarcadero, the GG Bridge, are pretty standard tourist fare, if that's what you want, but you asked what a local would recommend. Since you are European and in your 20's, I think that neighborhoods like the Mission, Hayes Valley, Russian Hill, and the like will be more lively and edgy and different for you, and more memorable than the cable car tourist zone. For a real experience, ride the N-Judah Muni streetcar out to Trouble Coffee near the ocean (Judah near 46 Ave), and chat for a while with the local guys who come and go.
I used to be a tourist, and now live here. I think Tim-n hits it right on the head when he points out your interests will help define the best area.
My first trips were all in the Union Square area and Financial District, but the neighborhoods are much more diverse in a number of ways.
San Francisco is really two cities in one -- a cosmopolitan city overlaid on a set of small villages, really.
I would avoid Fisherman's Wharf, Pier 39, etc if you want a sense of the living, breathing San Francisco.
On the other hand, I also agree that transport in the city gives you quick access to a lot of areas.
If, like Tim N and me, you like to mingle with real folk, or have special activities that define your tourist perspective, let us know and people can be helpful in pointing out what gets you best access.
What sets San Francisco apart from a lot of cities is our diversity -- in terms of population, food, and neighborhoods.
We are at the center of a food revolution that lets you eat fresh and interesting food at prices my European visitors find incredibly low, we have an explosion of food trucks and street food, we have a cults around differences in coffee, and baristas have rock star followings, we are experiencing a flowering of interesting, unusual, and downright peculiar artinasal ice cream. But you can reach those from anywhere in the city -- the question is taking a bit of time to decide what you care about and then checking up where they are and how to get there.
Also unusual is the diversity of Asian cuisines -- not just Chinese, but regional Chinese, and Vietnamese, Cambodian, Thai, etc.
You can spend all your meals seeing how these different cuisines treat fish.
But maybe food isn't your main travel attraction. What is?
