TITLE: What are the best topics to learn for a first (and second) course in Category Theory?
QUESTION [7 upvotes]: I am a mathematics student in my last year of undergraduate studies and I'm taking a first Course in Category Theory. The professor that is giving the course is not a category theorist and because of the different backgrounds of my classmates we are advancing slowly (we have almost finished the semester). I'm afraid we'll not cover the same topics that are studied in other Category Theory courses. So far we have seen the definitions for 

category, 
concrete category, 
subcategory, 
retraction, 
monomorphism, 
epimorphism, 
bimorphism, 
isomorphism, 
functor, 
natural transformation, 
product, 
coproduct, 
small category and 
quasi-category. 

I know we will eventually get to Yoneda's Lemma, so my questions are:

Are these topics enough for a first course in Category Theory for undergraduates?
If not, what are the best topics to learn in a first course in Category Theory for undergraduates?
I'm planning to take the following course, so I should better study ahead: What are. The best topics to learn in a second course in Category Theory?

REPLY [9 votes]: There are a few huge gaps in that list I can see (also quasicategories should be removed), namely

limits and colimits more general than products and coproducts, 
universal properties, and
adjoint functors. 

You should absolutely know what these are after spending a semester learning category theory because people (not just category theorists) use them all the time. Beyond that (e.g. for a second course), it depends a lot on your interests. Some random possibilities:

monads,
Kan extensions,
monoidal categories and 2-categories,
enriched categories and abelian categories,
cartesian closed categories and topoi.