You should have at least 2 costume slots (uniform and off duty). If you are part of a fleet, you will have 3 slots. To change into the different slots, just right click on your avatar, and select "Change uniform".
There may a keybind, but I don't know what that would be.
AFAIK, right now the extra slots are only for duty uniforms. The ability to make additional off-duty slots is supposedly being looked into.
I and a friend observed that if you buy two sets of slots, one of the second slot will be an universal slot after creating it and going back to modify. Not sure if that's by design but there you have it.