As a math graduate student, I second the choice for Abbott's "Understanding Calculus". It's a wonderful beginning book for analysis. Walter Rudin's "Principle's of Mathematical Analysis" is an amazing book but it's difficult to start with.

For a quick intro to Lebesgue integration you can read the beginning of Rudin's "Real and Complex Analysis" or Halsey Royden's "Real Analysis".

I haven't read Axler's book. I liked Hoffman and Kunze's "Linear Algebra"