The requirements to apply vary by town because towns add in random requirements to deter applicants even though they're not allowed to. Go to your local PD and get an application, then you'll know what's next. Be patient, law says the application must be approved/denied within 8 weeks, but many towns take much longer, but some process them quite fast.

Study CT gun laws very thoroughly. They are numerous, confusing, and illogical, and it's very easy to accidentally violate one and lose your license and guns forever. Don't ever bring a gun into NY or mass.