How to design such a Search Results Page? So, it shows Universities with courses based on two taxonomies chosen, discipline and degree level

How to design is pretty straightforward you just design it, and to build it you have to choose if you will use plugins to make all the custom steps so you are able to do this to handle filtering and displaying the courses, the other way is to make everything from the start with your own code and do all the necessary steps like build the taxonomies, post types, queries for the search the forms that are going to handle the parameters for the search, the functions that are going to display or retrieve data, etc