Because of my husband´s job, I have become a real pro at starting somewhere new (moved house 6 times in 10 years!).
I make a special effort to smile at people, and have a bit fo small talk (weather, something going on at school, try to remember what they tell me, and then when I meet them next and can ask how SO-and-SO went. Sounds like I am trying too hard maybe, but I think you have to try hard when you´re new).
I never assume cliques are "closed", most people like meeeting someone new (positive attitude), will always go up to someone to talk, even if I never talked to them before. I never stand by myself (even if it´s to ask: "Do you know what time assembly is tomorrow?" or "How did your DC get on with the project? We had a nightmare in our house making solar system out of plastic cups", "My DC lost his PE kit, where should I look for it you think?" etc. etc.)
I join the PTA and help with jobs like doing food or drink at school do´s (being the mulled wine person makes you very popular ). I helped shovel snow when the school was snowed in. etc. However, if I do not have time to help, I am sure people will understand, you don´t have to be a doormat
I invite kids that my DC like to our house, even if I don´t really know the mum. Ask the mum if she wants to hang around for a coffee (pick up is fine too).
It is very hit and miss, and I am aware that listing this I sound like I try too hard, but I think you have to really really try, no one will do it for you IYSWIM.
Bottom line: Everyone likes people with a positivie helpful attitude, who seem interested in other people.
PS, Last time I went to the soft play centre, I texted 6 people if they wanted to join me, 5 said no, but one came. Fair enough, and better than going alone.