I'm posting a separate answer because I realized my first might be too much algebraic topology. Another great second year course idea is to follow Lectures on Modules and Rings by T.Y. Lam. This book is my bible for homological algebra, and I have never heard anyone claim there was a better book for this. The only "downside" is that everything is non-commutative, but Lam does a great job of telling you exactly what commutativity gets you (via corollaries to the theorems), so even students who go on to work primarily in a commutative setting will not be ill-served.
If you wanted to cover lots of great homological algebra in this proposed second course (without an eye towards algebraic topology), I can't think of a better book. Here are some topics:
- Projective, Injective, and Flat Modules
- Semisimple, Coherent, Von Neumann Regular Rings, Cohen-Macaulay, and Gorenstein Rings
- Homological Dimensions and Regular Local Rings
- Quasi-Frobenius Rings and Algebras
- Matrix Rings and representation theory
Since this is the "second course" to his "First Course in Noncommutative Rings" one might be tempted to use that text for the first course. I'm not sure this is such a good idea. While I love Lam's writing style and the vast amount of material he covers, it seems a lot of what A First Course covers isn't really necessary to do algebra later on, i.e. a lot of it deals with situations which modern research avoids via standard assumptions on the rings in question.