I've been fumbling in slow motion through Professor Ghrist's Calculus MOOC on Coursera ( https://www.coursera.org/course/calcsing ), and while I find it challenging it's quite enjoyable. His presentation style is quirky and his manner of speech...unique, but he presents concepts in a very compelling and lucid fashion. It's obvious he and his team have put a tremendous amount of effort into the class, and it's especially apparent in the videos.
I enjoyed reading about what he does the rest of the time: unsurprisingly he is doing interesting things in applied math. Thanks for posting this kjak, I wouldn't have discovered it if it hadn't come up on HN!
I will try to list resources in a linear fashion, in a way that one naturally adds onto the previous (in terms of knowledge)
First things first, I assume you went to a highschool, so you don't have a need for a full pre-calculus course. This would assume you, at least intuitively, understand what a function is; you know what a polynomial is; what rational, imaginary, real and complex numbers are; you can solve any quadratic equation; you know the equation of a line (and of a circle) and you can find the point that intersects two lines; you know the perimiter, area and volume formulas for common geometrical shapes/bodies and you know trigonometry in a context of a triangle. Khan Academy website (or simple googling) is good to fill any gaps in this.
You would obviously start with calculus. Jim Fowlers Calculus 1 is an excellent first start if you don't know anything about the topic. Calculus: Single Variable https://www.coursera.org/course/calcsing is the more advanced version which I would strongly suggest, as it requires very little prerequisites and goes into some deeper practical issues.
By far the best resource for Linear Algebra is the MIT course taught by Gilbert Strang http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr... If you prefer to learn through programming, https://www.coursera.org/course/matrix might be better for you, though this is a somewhat lightweight course.
After this point you'd might want to review single variable calculus though a more analytical approach on MIT OCW http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab... as well as take your venture into multivariable calculus http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable...
Excellent book for single variable calculus (though in reality its a book in mathematical analysis) is Spivaks "Calculus" (depending on where you are, legally or illegally obtainable here http://libgen.org/ (as are the other books mentioned in this post)). A quick and dirty run through multivariable analysis is Spivaks "Calculus on Manifolds".
Another exellent book (that covers both single and multivar analysis) is Walter Rudins "Principles of Mathematical Analysis" (commonly referred to as "baby rudin" by mathematicians), though be warned, this is an advanced book. The author wont cradle you with superfluous explanations and you may encounter many examples of "magical math" (you are presented with a difficult problem and the solution is a clever idea that somebody magically pulled out of their ass in a strike of pure genius, making you feel like you would have never thought of it yourself and you should probably give up math forever. (Obviously don't, this is common in mathematics. Through time proofs get perfected until they reach a very elegant form, and are only presented that way, obscuring the decades/centuries of work that went into the making of that solution))
At this point you have all the necessery knowledge to start studying Differential Equations http://ocw.mit.edu/courses/mathematics/18-03sc-differential-...
If you have gone through the above, you already have all the knowledge you need to study the areas you mentioned in your post. However, if you are interested in further mathematics you can go through the following:
The actual first principles of mathematics are prepositional and first order logic. It would, however, (imo) not be natural to start your study of maths with it. Good resource is https://www.coursera.org/course/intrologic and possibly https://class.stanford.edu/courses/Philosophy/LPL/2014/about
For Abstract algebra and Complex analysis (two separate subjects) you could go through Saylors courses http://www.saylor.org/majors/mathematics/ (sorry, I didn't study these in english).
You would also want to find some resource to study Galois theory which would be a nice bridge between algebra and number theory. For number theory I recommend the book by G. H. Hardy
At some point in life you'd also want to go through Partial Differential Equations, and perhaps Numerical Analysis. I guess check them out on Saylor http://www.saylor.org/majors/mathematics/
Topology by Munkres (its a book)
Rudin's Functional Analysis (this is the "big/adult rudin")
Hatcher's Algebraic Topology
[LIFE AFTER MATH]
It is, I guess, natural for mathematicians to branch out into:
There are, literally, hundreds of courses on edX, Coursera and Udacity so take your pick. These are some of my favorites:
Artificial Intelligence https://www.edx.org/course/artificial-intelligence-uc-berkel...
Machine Learning https://www.coursera.org/course/ml
The 2+2 Princeton and Stanford Algorithms classes on Coursera
Discrete Optimization https://www.coursera.org/course/optimization
Some I can recommend that are still available on Coursera:
- Introduction to mathematical thinking 
- Introduction to Mathematical Philosophy 
- Machine Learning (actually a CS course, but involves linear algebra and some calculus) 
- Calculus: Single Variable 
Personally looking forward to the ebook coming out! I was actually just this morning going through and signing up to a bunch of mathematics related Coursera courses. For those interested, quite a few are starting soon:
[Jan 7th] Calculus: Single Variable - https://www.coursera.org/course/calcsing
[Jan 7th] Calculus One - https://www.coursera.org/course/calc1
[Jan 28th] Algebra - https://www.coursera.org/course/algebra
[Jan 28th] Pre-Calculus - https://www.coursera.org/course/precalculus