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If you want to be serious about math you should get a feel for what mathematics means to mathematicians.

A common view is that mathematicians prove theorems. While true, a skill that is not emphasized enough is learning (by heart) and understanding definitions. Learning by heart here also means something slightly different than simply being able to recite definitions and theorems. You have to be able to compare the objects that you define and get a feel for how a definition is really a manipulation of a basic intuition.

A great book to start with is Rudin's Principles of mathematical analysis. You will get a much deeper appreciation of what calculus is. If you want any chance of understanding the mathematical tools used in theoretical physics (operators, Hilbert spaces, Fourier decompositions to get solutions to differential equations etc.) you will have to understand analysis.