I retired about 9 years ago from the Air Force. I live in far west San Antonio (outside Loop 1604) and go to either Lackland AFB or Ft Sam Houston (near central San Antonio). Both are less than 20 minutes away and are larger bases than Randolph AFB. After you retire and assimilate into the civilian community, going to the bases will be less frequent, so living close by should not be a concern.
Here are some links to the "other" activities you are looking for.
Good Luck and welcome to San Antonio!
http://www.sportstavern.com/texas/sanantonio.html
http://www.samustang.com/index.html (broken link)