Ugh! This has been the story of my life since I had a child. Combination of most of my good friends moving away and the ones here, well I guess losing contact as most don't have kids or kids not the same age. I got so depressed about my lack of friends that I decided with gusto one day to really reach out to other moms who seemed cool. I said to myself, can't be worse than mustering up the courage to ask a guy out right? Well, think twice before you do that. I met one mom one day with a gorgeous son exactly my son's age and she seemed so nice so I gave her my number. She sounded like she would be into playdates and I thought she'd call, but NO! I felt so rejected. It actually made my feeling sorry for myself and loneliness even worse, and made it harder to make friends for a while. I kind of gave up (not just because that one incident).
Eventually I started bringing my son to this one playground he really loves and there are always a lot of kids there his age. It's not the most convenient one to get to, but I don't care for my neighborhood park much. I had no intention at that point of making friends. It turned out there were a group of moms and dads who regularly brought their kids there around the same time on most days. It took all summer but eventually I started feeling like we are friends. They are all great and I love their kids. That's my advice to you because then you can build up friendships over time, you'll know if the friend vibe isn't happening. In the meantime, some of the playground chat with other parents helps fulfill the need for true friendship. Like for me, before I really clicked with anyone, at least I was connecting with some other parents and able to talk to people about parenting issues. If you find someone there who you like a lot, you can ask if they come on certain days, or if they are there regularly, without having to arrange a formal playdate. Takes the pressure off them too, they don't feel like they have to call a total stranger, but if they are thinking of going to the playground and you will be there, they may look forward to it more.
Sorry this is so long but it is such a dear personal issue for me right now and over the past 2 years, I really feel for you!! It still hasn't all worked out for me either, and with winter coming, soon it will be dark before I get out of work. I'm hoping my new friends want to stay in touch over these cold months as much as I do. If not, I'll probably be sad and posting a question like yours! :( Good luck to you