Consider a folding bike for multi-modal travel - you'll have relatively unrestricted access to the good public transportation network (like Muni trains and buses, rush hours on the BART, passenger only ferries, taxis/Uber, etc.), and the bike is easier to store/take inside (eg, rainy season/theft concerns). Make a cheap nylon cover for it, and turn a bike into 'luggage.' You could use Muni public transport for an uphill section and then ride a long gradual downhill; shoot out to different part of town to explore, or BART out to the burbs where I've seen very scenic MUPs. $150 is not a lot of money, but doable in the used market.

for $150, you pretty much have to go used, since any quality new bike will go for $400 and more.

At 5'10", you should be fit a common size, maybe 19 to 20" mountain bike or hybrid, M/L to L, or 55 or 56 cm road bike. A lot depends on you proportions, like whether your height is in your legs or your torso, so make sure you test ride for fit.

If if I were looking for a used city bike, I would look for low gears, triple chainrings, 7 to 9 speed cassette. It should have braze onsto fit a rack, and clearance for fenders. Frame material could be cro moly steel or aluminum. I would suggest a used hybrid from the 90s to a current model. Brands to look for? Lots of choices. Trek, Bianchi, giant, Kona, Specialized, Marin, Raleigh, Jamis, Fuji, Surly are some can't miss choices. Other brands like Schwinn, Diamondback, and Mongoose were good brands, but some models are cheap, department store bikes to be avoided. But if you can tell the difference, worth getting these bikes. This is not an exhaustive list. There are other small brands well worth considering.

Also, pay attention to condition. A $250 used bike fully restored and ready to ride is a better value than a $150 bike in rough shape.

There are several bike tour companies in San Francisco. They’ve outfitted a lot of people with bikes and have gotten lots of feedback on the results of their choices. They’ve also had to make decisions about what bikes to invest their money in.

It might be interesting to have a conversation with one or more of the tour companies. While you’re at it, you could rent a bike for an hour to see if you like it. Wheel Fun Rentals is one such company I used when I was in town. They had a range of different bikes available for rent. A similar conversation with a local bike shop that also rents bikes might be similarly productive. Who knows, but you might find that any of them sell off their rentals for a good price after a certain amount of use.

-You'll want a triple chainring (3 front gears) to get low climbing gears
-Probably want a hybrid (a "normal" bike to non bike people)
-Lighter is better for climbing so I would seriously consider upping the budget to $400-$500 (why spend money twice to get something better later?) and get a proper bike shop bike which will be 20-25 lbs

A base specialized Sirrus or Trex FX would be a great choice.

https://www.specialized.com/us/en/me...=239681-129091

https://www.trekbikes.com/us/en_US/b...colorCode=grey

You should build your own bike. I built mine for 900 and it weighs 13.5lbs... I built my sisters for $400 and it weighs 19lbs both are carbon fiber

People that don’t plan on seriously pursuing something usually don’t take the the time to register and post on a specific forum.
Not a criticism. Just a heads up that maybe you are a tad bit more serious than you believe which means that you may want to spend more than the cost of a concert ticket or snifter of Louis XIII on a bicycle.

SF has a big fixed gear scene. A geared bike is by no means a necessity unless you you live in the steepest areas of the sunset, for example. I did, and some spots are steep enough you have to walk even with gears. In fact, fixed or ss is an advantage in some spots as you don’t have to worry about chain slip on a climb, but that’s just imo. I’m not advocating for riding fixed, merely suggesting that it is quite common.

Ultimately, the best choice will come down to where you live, how you will ride, budget and theft tolerance. Get there, find a place and then check out CL, or any other very many bike shops. There’s no need to buy something in advance. The market is so active that you will have no issue buying a decent bike the same day you start to look.