In what order should the following areas of mathematics be learned?

I'm unclear what you mean by "Algebra"; if you mean stuff like working with polynomials, basic equations, symbolic manipulation, etc., then that goes first. If you mean "abstract algebra", then you can wait.

Added. Likewise: if by "geometry" you mean classical geometry, or even projective geometry, then the following applies.

Calculus, Discrete Mathematics, and Geometry, are independent enough that their order doesn't matter.

Added. However, if by "geometry" you mean analytic geometry, then it should definitely precede calculus, and the same is true if it means trigonometry. I think it unlikely that you meant "differential geometry" or "algebraic geometry", but if you did those are very advanced topics that should wait until well after calculus, abstract algebra, and real/complex analysis.

For introductory probability and statistics you'll find Discrete Mathematics very useful; for more advanced probability and statistics, Calculus is a must.

An "introduction to proofs", which would include some set theory, some basic logic, etc., can be done at the same time as Discrete Mathematics, or immediately after.

After all this, then you can hit linear algebra, abstract algebra, real or complex analysis, in pretty much any order (though complex analysis should follow real). Abstract algebra is a bit easier if you've taken linear algebra, but this is not strictly necessary.

If you happen to find probability and statistics very interesting, then you should do some measure theory after the real analysis.

This is the order I would suggest:

• Linear Algebra
• Calculus
• Discrete Math
• Probability and Statistics
• Geometry

The first two are interchangeable. According to some people(article about teaching math), discrete math should really be taught first.

In my opinion you want to learn the topics in this order :

• Algebra
• Geometry
• Calculus/linear algebra, (statistics if you want, not necesarry)
• real analysis
• complex analysis

Any duplicates on a line means that you can learn them simultaneously! I would assume that you would cover some calculus rigorously in analysis and also touch upon some set theory, and of course, learn the fundamentals of "proofs."

I'm doing the same thing.

I find its best to interleave your study of these topics.

1   algebra / geometry / calculus          | math
2   discrete math                          | software pkgs
3   probability+stats                      | (matlab/octave/maple)

You'll find a math software package takes a while to get used to (esp if you don't have a programming background). So starting with one right away is a good idea. Octave is a free version of MATLAB. Be warned: Octave can be painful to use at times. MATLAB is the Cadillac.

Also if you're doing proof-based study, you'll find a lot of calculus and algebra (at least as it appears to me at the moment) is mostly orthogonal. That is, the Fundamental Theorem of Calculus won't really help you to prove the Triangle inequality. But how to prove and your way of thinking should be exercised by proving in either topic.

Algebra is closely related to geometry. Linear transformations, geometric intersection - to me geometry seems to boil down to simply applied algebra.