In order:

1. Book of Proof

1a. How to Prove It: A Structured Approach

2. Elements of Set Theory - Enderton

2a. Naive Set Theory - Halmos

3. Foundations of Analysis - Landau

4. Basic Mathematics - Lang

5. Principles of Mathematics - Allendoerfer & Oakley

6. A Transition to Advanced Mathematics

6a. Concept of Modern Mathematics - Stewart

7. Calculus - Apostol or Spivak

Don't skip any