J.R.
Join a moms' group (you can find a bunch on Meetup.com). A lot of times, they organize playdates at local parks or people's houses. It's a nice way to make friends for you and your child(ren).
Balboa Park has free museum Tuesdays. Each week, a rotating group of museums offers free admission to the public on Tuesdays. The complete free Tuesday schedule is here: http://balboapark.org/calendar/detail.php?EventID=370
There are a bunch of playgrounds around San Diego you can check out. There's a list here: http://sdplaygrounds.com/
The San Diego Continuing Education system offers free indoor and outdoor classes for children up to five years old and their parents. I don't think the spring term schedule is out yet, but you can find more information on the classes here: http://www.sdce.edu/class-schedule/class-schedule.php (You'll want the ones under "Child Development.")
Welcome to San Diego! :)