Solutions To Advanced Calculus Second Edition Fitzpatrick.zip
It sounds like you might simply want to look at an advanced calculus book which starts from an axiomatic point of view. I like Fitzpatrick's "Advanced Calculus" which starts from the axioms for the real numbers, and Tao's "Analysis I" which starts from the Peano axioms for the natural numbers.
solutions to advanced calculus second edition fitzpatrick.zip
But I don't know of any such book which assumes that the reader has never seen calculus. You could simply pair an advanced calculus book with a standard undergraduate calculus textbook, dipping into the latter as needed whenever the more advanced book moves into a new topic. You'll certainly find many more elementary exercises in a standard textbook, which might be useful.