There is not much to do in Palo Alto outside of eating and going to like 2 or 3 bars. Palo Alto is also incredibly expensive so most of your activities will boil down to eating out or drinking, which is difficult to do often on a student budget. The surrounding areas are gorgeous though. If you enjoy hiking/nature there are amazing trails less than half an hour away from campus. Most trails require you to drive to them, so if you have a car or hiking enthusiast buddy with one then you are set. The city (San Francisco) is also a 45 min - 1 hour train ride/drive away depending on which train you take or time of the day. The issue is that, just like Palo Alto, San Francisco is incredibly expensive and everything aside from walking around or hanging out in Dolores Park requires money. San Jose is a bit closer (40 - 30 minutes drive, 45 train), but there isn't much to do there other than eating/drinking.
I think the difficult part is getting off campus, even if you are just going to Palo Alto to eat/drink. Campus is massive and it definitely has a bubble effect that keeps many people in. This is mostly due to most students (grad and undergrad) living on campus and honestly campus having lots of activities you can do (2 gyms, pretty to walk around, like 4 museums, a hiking trail nearby, etc...). Can't talk much about sporting events because I don't like attending them, but as a Stanford student you get tickets for free (or VERY reduced price) for virtually anything happening on campus.
