Books that:
has an introduction to proof, logic, and topics like sets and groups. Books
can prepare you for rigorous calculus texts like Spivak and Apostol.
Question 1. I've been looking at the books by Gelfand (Algebra, Trigonometry, Functions and Graphs, The Method of Coordinates). I'm not sure if they cover all of the precalculus curriculum, though.
Question 2. What would be some good calculus books other than Spivak and Apostol?
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$\begingroup$IMHO, you are not gonna find a truly amazing pre-calculus book. Just read pre-calculus topics from random sources. You can consider these books:
$1$ George F. Simmons-Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry
$2$ Michael Sullivan III -Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
For a bit higher level, consider these.
Gilbert Strang-Introduction to linear algebra.
Amann, Herbert, Escher, Joachim- Analysis $1$.
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