While lower bike weight and lighter wheels can help, in reality you are the one who has to climb the hills and go fast. Generally a good climber can also go fast.
Since no one knows your ability no one can say what the best climbing fast bike will be.
As for shoes, if you have the legs, you can wear whatever you want.
That said using regular bike and ebike in the same sentence may not be a good indicator of your riding experience.
John