TITLE: What is the best sequence in which to study undergraduate mathematics?
QUESTION [10 upvotes]: I am currently self-teaching mathematics at the moment, with my current focus being pre-calculus mathematics. At some point I will eventually reach undergraduate material and as I do not possess a great grasp of the subject I would like your thoughts on what would pedagogically speaking be the best sequence in which to study the numerous topics that compose mathematics.
It is very obvious to me that in order to study certain topics some pre-requisites are necessary. I am certain many of you have studied mathematics and with the power of hindsight may have wished you had studied a particular topic before/after another as it would have perhaps enhanced your understanding and/or made certain concepts easier to digest, I'd like to benefit from your experience.
Making my query more pointed: I would like to know the ideal sequence in which to self-study mathematics, along with canonical book recommendations for each topic.

REPLY [3 votes]: I would recommend these to be read at the same time: 

Euclid's ELEMENTS (translated by Heath), and 
Hartshorne's GEOMETRY: EUCLID AND BEYOND. 

Proving theorems by construction is a wonderful way to do mathematics, and Hartshorne's book will point out and correct the logical mistakes Euclid made, extend his results, and provide a connection to contemporary mathematics by way of introductory modern algebra. Best of all, the prerequisites are zero, zilch, nada. By the time you have finished both of these books (which should take the better part of two semesters going it alone) you will know what you want to know more about.