Computation Complexity books for a mathematician
Michael Sipser's text book Introduction to the Theory of
Computation
is a classic introduction to computation theory, and gives an
introduction to complexity theory in the end.
I don't know how to answer the question better than just providing the
table of contents of the book.
PART 1: AUTOMATA AND LANGUAGES.
1. Regular Languages.
2. Context-Free Languages.

PART 2: COMPUTABILITY THEORY.
3. The Church-Turing Thesis.
4. Decidability.
5. Reducibility.
6. Advanced Topics in Computability Theory.

PART 3: COMPLEXITY THEORY.
7. Time Complexity.
8. Space Complexity.
9. Intractability.
10. Advanced Topics in Complexity Theory.