I think category theory can help to give a "roadmap" for what the basic theorems of each course will be. Personally, as a student, I always felt like my courses jumped around too much without explanation, and like I could never see what was coming next. Knowing category theory has helped me see the bigger picture. When we encounter a new category, it helps to understand what products, limits, colimits, functors, natural transformations, Yoneda Lemma, adjunctions, etc. look like for this category. Indeed, most of the theorems in a topology or algebra or measure theory course fit into this mold -- we define what a morphism is and then prove a bunch of facts about products, functors, etc..
We should teach the bulk of Lawvere theories only once. Students should get to know groups and rings as two flavors of a similar construction, and also be introduced to the mysterious fact that some objects, like fields, don't fit into the Lawvere-theory paradigm.
