I've had to start over with friendships many times due to moving often. One thing I eventually learnt was that not everyone I met had to become a friend. I realised that it was perfectly fine to have people I know and then have a deeper relationship with others who became close friends.
I also learnt that no one is waiting to be my friend and I had to put in the spade work. There are the obvious things to do. such as mixing with others so you have a chance to meet that one person with whom you hit it off. Showing interest in the other person is always good, so have a few polite questions ready e.g. 'Do you live locally?' Or 'I love your hair, where do you get it done?' or whatever's appropriate to the situation.
Then, and I think this is crucial, if you get the feeling you're on the same wavelength, try to make a firm date to get together. Ask if they'd like to meet for coffee next Wednesday - not at some nameless point in the future. They may jump at the invitation, or they may say, 'Oh, I can't do Wednesday but Thursday would be good,' or they may say, 'Sorry, I can't manage that. I'm busy.' The latter response can be a bit ouch but you can rationalise it as nothing ventured, nothing gained.
Don't forget to offer people a way of contacting you, too. (Use a 'spare' email address if you're worried about security.) If they reciprocate, that's even better.