I used meetup and joined two local groups. Those groups are big and they post lots of play dates hosted by members of the group as well as outings to parks, farms, etc. The first date I went to, the moms all had toddler boys, I had an infant girl, usually there are age ranges posted though. The next I went to was a bunch of kids around DDs age 4 months and the host said she preferred to have a set group instead if never knowing who would show. We agreed. 2 years later we have been meeting once a week, we go to each others birthday parties both ours and the kids, and we have a FB group, aptly named Tuesdays with mommies, where we chat during the week and make our plans. Best of all, four of us are on round two, making the next generation of play group.
I also was very nervous and shy about making friends. But as my only peer to be married, let alone have kids, then to become a stay at home mom, I needed them.
I love my group of six moms and their adorable children. Sorry this is verbose! Good luck, I hope you find someone as wonderful!