I've long believed that the key determinant in making new friends is how often you see someone.
In other words - a normal friendship pattern is
A meets B in group setting.
A and B keep meeting in group setting.
A and B find they like each other a bit/have potential "friendship chemistry".
Typically A and B do something socially still in the group setting (eg. if at work, colleagues go for a drink; if at the school gates, parents going for a coffee) and confirm chemistry.
Then one of them asks the other to "do something" together - whether that is grab a coffee, have lunch, go to an art gallery they were both interested in or try out the new wine bar that has just opened - matters not.
From that a friendship is born.
The key to it is regular contact with a group of people - because if you met a 100 people, you'd probably only want to be good friends with one or two.
You can get this in lots of places - work is the obvious example, regular volunteering, going to the same place every day at the same time (pub/sandwich shop), joining an educational course that takes a term or so, joining a regular activity club - bell ringers etc.