You cannot go to every shop as a visitor without getting a membership. check the website for the location where you want to go and look if the have a visitor (ビジター) package.
Depending on the location, they might require you to speak Japanese. Since Gym facilities have a legal responsibility towards your health, they have to make sure that they explain everything to you, including the machines etc. If they cannot do so in Japanese, they will either have to have English-speaking staff or simply will refuse you to enter. That means also that you will have to stick to the rules they tell you. If you are used to do things differently in the USA than what they tell you, better forget about what you know and do as they say, specially if you want to be a walk-in guest.
You will have to bring spare shoes. They will not let you train in your street shoes. All other facilities are depending on the location normally.
The chance that something will get stolen from your locker in Japan is very, very, very low. They rather come running behind you if you forget something.