Here's my list:
Basic
1. Chapter Zero (Carol Schumacher) 2. Naive Set Theory (Paul Halmos)
Linear Algebra
1. Finite-Dimensional Vector Space (Paul Halmos) 2. Linear Algebra Done Right (Steven Axler)
Real Analysis
1. Real Mathematical Analysis (Charles Pugh) 2. Introduction to Analysis (Maxwell Rosenlicht)
Algebra
1. A first course in abstract algebra (John Fraleigh)
These books are better for self-directed reading compared to some of the classics like Rudin or Herstein. These should keep you busy for a while.