I've just started reading a volume of books by Hung-Hsi Wu[0]. They are main audience is teachers of mathematics. Two other books that seem to offer a formal rigorous approach of teaching basic math are Principles of Mathematics by Carl Allendoerfer and Modern Algebra - A logical approach by Pearson, H. R and Allen, F. B

Needless to say, these books are focussed on High School math but should be sufficient preparation for courses on a more rigorous level.

My email is in my profile if you are looking for somebody to study with.

[0]https://math.berkeley.edu/~wu/