What would be a good way for me to go about volunteering in a childrens hospital?

Call. In my professional experience, every hospital has had a volunteer coordinator who oversees the volunteer programs. Volunteers usually have to be screened and undergo some training once (and if) chosen. It is a beautiful thing for a person to do and i strongly suggest you call the hospital and explore how their programs work.
Just Ask. Most hospitals have volunteers that help them, especially with kids. You can call the hospital and ask the operator to put you in touch with the office responsible for volunteers there. Many hospitals will require that you go through classes or training before volunteering. Volunteering at a children's hospital can be a valuable and rewarding experience so enjoy!