Add fieldset with add more button using ajax
This is easy to achieve with a content type and the Field collection module.  With that you can set up a qualification field of type field collection, then go to admin->structure->field collections and add the individual fields (School / College / Course).  If the qualifications field has a "Number of values" setting of "unlinmited" then you'll automatically get the "Add more" functionality as you described.

You may need to also use Field Group or Field Collection Fieldset in order to make the fields appear in a fieldset.