>>110954451. It's a useful language for expressing basic algebra things like the universal properties in the First Isomorphism Theorem, in a way that works equally well for groups, rings, or other suitable structures. I.e. it performs the same function as Universal Algebra.
2. It's a useful language for some kinds of algebraic topology.
3. You can study it directly, and get some interesting questions and theorems about categories themselves. However it's important to be very good at 1 and 2 before you try 3. Otherwise you will not be able to discern interesting from uninteresting (or already solved) questions, and worse, will not be able to communicate your ideas to other mathematicians.
