Welcome to San Francisco! Some ideas for specifically what you said you’re looking for (many of these restaurants have been mentioned already in this thread):
Authentic Mexican place:
Note that burritos are not really “authentic Mexican” but rather “authentic” Mexican-American, but perhaps try a Mission-style burrito which was invented here in San Francisco’s Mission District. A cylindrical flour tortilla tube filled with rice, beans, salsa, and protein, and optionally guacamole, sour cream and cheese. Like Chipotle but better (usually). Some good places to get one are El Farolito on 24th and Mission and La Taqueria near 25th and Mission. La Taqueria has a slightly non-traditional and slightly less filling burrito as they don’t put rice in it. Not that it means much but La Taqueria’s burrito won best burrito in fivethirtyeight’s burrito bracket. We have some burrito threads on HO too.
La Torta Gorda is indeed good if you’re looking for a torta (try the rajas con queso torta), as is Tortas Los Picudos down the street. That street (24th Street in the Mission District) has many good Mexican restaurants, and is also home to La Palma Mexicatessen, a Mexican deli that also makes a good burrito along with other items like gorditas.
Tommy’s Mexican Restaurant was also mentioned and also falls under SF institution. Definitely old school, can’t vouch for authenticity though. It’s where the Tommy’s Margarita was invented. It’s also somewhat near Golden Gate Park if you happen to visit the park.
Local restaurant institutions:
Some old institutions:
House of Prime Rib is a great old school prime rib restaurant. Try to get reservations if you consider going, they also take walk-ins at the bar.
Swan Oyster Depot which you mentioned, is worth going to. Prepare to wait if you go though! Also bring cash. I like the Sicilian sashimi and the crab backs. And also the oysters and clams of course.
Tadich Grill qualifies as an institution. It is the oldest continuously run restaurant in California (though not at the same location). Stiff martinis, sand dabs, and cioppino. Served with a hunk of San Francisco sourdough.
Tosca Cafe in North Beach (our Italian-American neighborhood) is over 100 years old, which in the US makes it an institution. It only started serving food relatively recently though. Italian-American cuisine along with a famous “cappucino” that doesn’t contain coffee.
If you happen to be around Fisherman’s Wharf doing touristy things, try an Irish Coffee from the Buena Vista Cafe. I can’t speak for the food there though.
Some new(er) institutions:
Zuni Cafe is definitely worth visiting. Cal-American cuisine. Famous for its roast chicken with bread salad.
Nopa also Cal-American. Great burger & fries and pork chop.
Delfina - Cal-Italian, should be opening back up its dining room soon. Not sure if you’re interested in Italian food so much though coming from Europe.
Perhaps check out the farmers’ market in The Ferry Building since you are here on a Saturday. The Ferry Building itself is worth checking out as there are many food vendors within. If you’re in the mood for seafood in the Ferry Building I recommend Hog Island Oyster Co.
