Hola!
Supervisor=el/la superintendente
Manager=el gerente/la gerente/el jefe/la jefa
Facility=la falcultad
Maintainence=mantenimiento
Please get me the supervisor=Por favor, agarrame el superintendente (Be careful with this. Usually in a business environment you don't use commands. It would sound better to say "¿Me agarrra(s) al superintendente, por favor? or "¿Me puede(s) agarrar al superindente, por favor? Can you get me the supervisor please?
I am looking for the supervisor=(Le) estoy buscando al superintendente/(Le) estoy buscando a la superintendente (this one is if it's a woman)
I need to speak with the supervisor=Necesito hablar con el superintendente/Necesito hablar con la superintendente
Your dad would probably want to talk to his supervisors and to his coworkers, that he doen't know very well, in the "Ud." form. Usually, you talk to costomers in the U.d form and to people that you don't know very well. Also, you use it to show respect to your maneger. The "tu" form is what friends use and it is used among coworkers. Sometimes, kids talk to their parents in the ud. form in some Spanish speaking countries, but usually they talk in the tu form. When in doubt I think it's always best to talk in the ud. form. It's not insulting to be too formal with someone. It would only sound weird if a kid talked to another kid in the ud. form.
Espero haberte ayudado
Saludos