> 4) Calculus and Linear Algebra are the father and mother of applied math.
> You'll save yourself a ton of grief if you learn them first
Baby Rudin and Axler are used currently by Harvard Math 55 to teach those subjects. Rudin might not be very didactic (I would be happy to hear about alternatives), but Axler is a fantastic choice.
If you liked Axler, you might check out Abbott's Understanding Analysis, also in the Springer UTM series. I think it covers somewhat less than Rudin (e.g. looking at baby Rudin's contents, I'm pretty sure Abbott doesn't touch Lebesgue integration) but it's a pretty great introductory analysis book IMO.
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"
> You'll save yourself a ton of grief if you learn them first
Baby Rudin and Axler are used currently by Harvard Math 55 to teach those subjects. Rudin might not be very didactic (I would be happy to hear about alternatives), but Axler is a fantastic choice.