Austinite here with some San Antonio experience.
Local in Austin could have any of several meanings... real people here include artists, techies, social workers, entrepreneurs, comedians, waitstaff, musicians, students, political activists. Natives are rare and include all of the above. The normal Austinite is from somewhere else, we don't breed that fast.
One experience that pulls the comedy-creativity-and-politics threads into a fun package is Esther's Follies. Originally a sketch comedy show that dipped into politics along with various local cultural stereotypes, it is now an institution that mixes current events jokes with popular gags that have lasted in some cases decades. A fun entry point, though costly ($15-20/person). Cheaper option: the Velveeta Room next door for free to $5 depending on what night. If comedy is your thing, maybe Coldtowne Theater - real working comedy club, good quality.
For my money, a show at a local innovative theater space such as Salvage Vanguard Theater is worth visiting because you don't find that in a small town. Thursday shows sometimes have a pay-what-you-choose option.
A friend of mine invites me occasionally to a little bar that plays live music and has real live plain old neighborhood people (often black) as clientele, with a fire pit out back. Skylark Lounge. Last month Yelp somehow rated it #1 local bar in America based on reviews. No idea if it's been overrun with Inauthentic Hipsters as a result.
One place to get affordable eats is the local dominant grocery store, HEB. Some of them have prepared food as well as groceries, and an air conditioned space with wifi where there are tables to sit at, relax, plan your next move. If you're stuck downtown in the hurly-burly, you can do the same thing at the actual national flagship store of Whole Foods (obviously the food is more expensive). People watching at HEB is more regular people, Whole Foods more of All the Downtown People in their daily glory. In summer on I think Wednesdays you can join free yoga on the roof of Whole Foods at 8:00 pm. I know truck drivers who do yoga, maybe that counts as real people here?? :) By the way, the same rooftop space sometimes has free theater by local troupes who like to put on a community show. People bring their own chairs and blankets, you could strike up a conversation.
Hmm... if you want to meet up one night or day, I could join/host some events for the day. I'm a local! PM me if interested.
Re San Antonio, I have friends there whose opinion is that local means dining at one of the many "Mexican" (Tex Mex) restaurants there. Tons of authentic ones. Cheap, delicious. Austin has been getting nose-in-the-air about its great trendy food, but get tacos and enchiladas and such as several local restaurants in San Antonio! Have "puffy tacos" at least once. :)
I second the motion re Natural Bridge Caverns. Very cool!
In between the cities near a town called San Marcos is a river wildlife research area. You can tour the exhibits in the main building for free, and take a fairly inexpensive tour in a glass bottomed boat. Very calming and worthwhile.
http://www.meadowscenter.txstate.edu/