Outside of BART, I didn't find that they public transportation was anything great. You can use the cable cars to get around some areas. If you want to go someplace like BART and the cable cars don't go, it might be easier to take a cab.
Check you smart phone apps. The standard map app that comes with the iPhone is really helpful planning navigating an unfamiliar place driving, walking or using public transit.
You'd be better off staying in the South Bay if you're concerned about costs (take your pick, anywhere from Los Gatos to San Jose to Palo Alto - all 90 minutes or less outside of the city, and plenty of stuff to do in that stretch of cities) or just straight up heading toward Napa or over to the coast and going north or south and making one of those places your destination while SF becomes the side-attraction. The Bay Area isn't a cheap area in and of itself but South Bay tends to be cheaper than San Francisco. You can easily do the tourist things in San Francisco in two, three days max. The rest of the Bay Area (plus the coast and Napa) is worth exploring far more than SF, IMO. There's several museums and other attractions in the San Jose area, many nice small-town down town areas in Los Gatos, Mountain View, and Palo Alto. Stanford is an interesting campus to see. Out on the coast, Santa Cruz (though somewhat touristy in areas) and Monterey are nice larger beach cities. Go north of SF and you've got Napa, Sonoma, and the Russian River. There are tons of great hiking opportunities all over the Bay Area.
Public transport is okay in SF if Bart is near where you want to go, or you don't mind waiting for a bus. Otherwise its walking or cabs. If you're staying outside of the city but on the peninsula, its real easy to take CalTrain up to the city then walk or catch a cab elsewhere. Driving in the city is the shits but it can be done. Its just a huge stress and a huge PITA.
As for attractions, there's various museums and stuff in the city, like art museums and the Comic museum (which is slightly underwhelming and not much cape 'n' tights if that's what you're mostly into, though I think right now they have a special Image Comics gallery). Yerba Buena Gardens is somewhat interesting. Check out all of the museums and lovely naturey things in Golden Gate Park like the science museum thing and the Japanese Tea Gardens. The zoo is pretty nice. Union Square as a destination itself isn't very exciting unless you've never been in the middle of any big city, ever. Fisherman's Wharf is super touristy but its something to do. Coit Tower is nearby but a bitch to walk to if you can't handle some steep hills. Chinatown and the other various ethnic-towns all have their quirks but I'm not sure they're much more special than other big cities that have ethnic enclave type areas. The Presido is interesting to be in but not a ton to really see or do. Its right next to the Golden Gate Bridge which is neat to drive or walk across. The hills directly on the other side of the Golden Gate Bridge provide some nice hiking opportunities, or just nice viewpoints to drive up to and look out from.
You can get around pretty well on the Muni buses if BART doesn't cover all destinations. It's probably a lot better than you're used to in PDX. I would suggest Alcatraz or Steinhart Aquarium, my Dad liked those even when he was in his 70s and couldn't walk around as well as before.
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