K.B.
Join a MOPS (Mother's of Pre-schoolers) group. They meet at local churches. They have regular meetings with childcare, organize playgroups and outings and even babysitting co-ops. Go to their website, mops.org and search for groups near you. I know there is a great group at my church, Wilshire Baptist, which is at Mockingbird and Abrams. That is fairly close to you, but you might find another group closer or at a church of your faith.
When my kids were little, I lived near NorthPark mall and it was one of my favorite places to go when it was too hot or cold to be outside. I pushed them around the mall in a stroller. There used to be an FAO Schwarz toy store where my son would play with the trains. Not sure what toy stores are there now, but I think there is a Lego store. There is a very nice park off of Mockingbird on Williamson, Lakewood Park that has some shade. Also a nice park on Hillcrest south of Northwest Hwy.