What sequence should I study these topics in? I will shortly list a series of topics that amount to what is essentially the first two years of an undergraduate degree. 
I'd like to know what is considered best order in which to study these topics. I understand that some of these are related topics i.e. Analysis, so if like topics could be grouped together that would be appreciated. Any other suggestions that you feel are relevant would be great too. 

Topics
1. Groups: Examples of groups, Lagrange's theorem, Group actions, Quotient groups, Matrix groups, Permutations.
2. Vectors and Matrices: Complex numbers, Vectors, Matrices, Eigenvalues and Eigenvectors.
3. Numbers and Sets: Introduction to number systems and logic, Sets relations and functions, the integers, elementary number theory, the real numbers, countability and uncountability.
4. Differential Equations: Basic calculus, first-order linear differential equations, nonlinear first-order equations, higher-order linear differential equations, multivariate functions: applications.
5. Analysis I: Limits and convergence, continuity, differentiability, power series, integration.
6. Probability: Basic concepts, axiomatic approach, discrete random variables, continuous random variables, inequalities and limits.
7. Vector Calculus: Curves in $R^3$, integration in $R^2$ and $R^3$, vector operators, integration theorems, Laplace's equation, Cartesian tensors in $R^3$.
8. Linear Algebra: self explanatory
9. Groups, Rings & Modules: self explanatory
10. Analysis II: Uniform convergence, uniform continuity and integration, $R^n$ as normed spaced, differentiation from $R^m$ to $R^n$, metric spaces, the Contraction Mapping theorem
11. Metric & Topological Spaces: Metrics, topology, connectedness, compactness
12. Complex Analysis: Analytic functions, contour integration and Cauchy's theorem, expanions and singularities, the residue theorem. 
13. Complex Methods: Analytic functions, contour integration and Cauchy's theorem, Residue calculus, Fourier and Laplace transforms.
14. Geometry: Groups of rigid motions of Euclidean space, spherical geometry, Riemannian metrics, embedded surfaces in $R^3$, length and energy, the second fundamental form and Gaussian curvature.
15. Variational Principles: Stationary points for functions $R^n$, Functional derivatives, Fermat's principle, second variation for functionals.
16. Methods: Self-adjoint ODEs, PDEs on bounded domains, Inhomogenous ODEs, Fourier transforms.
17. Numerical Analysis: Polynomial approximation, computation of ordinary differential equations, systems of equations and least squares calculations.
18. Statistics: Estimation, hypothesis testing, linear models.
 A: I suggest the following sequence:
3-5-2-8-4-7-1-9-10-6-18-11-14-12-13-16-15-17.
Exact contents and prerequisites of your material may call for some permutations, but this shouldn't be too bad.
The list of topics seems good to me.
As you proceed, you will develop a taste that will tell you what you want to focus on and what requires clarification.
When reading a book, don't hesitate to decide that you need to study some other theory first before you can lay hands on what you want to do.
What you need is a good start, and that looks like one.
A: You pretty much need #3 and #7 before anything.  Which order you do these in doesn't matter. A lot of stuff on your list will require multivariate calculus, which is why I say #7.  And in #3 you will build up your mathematical maturity/proof skills.
Then a whole world will open up to you, so you could do whatever you find interesting. I wouldn't recommend trying to plan much beyond that because you'll probably change your mind later anyway.  
9 comes after 1. (EDIT: I would study this after Linear Algebra/Complex Numbers because many of your groups will come from Matrices/Complex Field)
8 comes after 2.
17 and 16 come after 4.
10 and 11 come after 5.
13 comes after 12
14 whenever.
18 comes after 6.
A: Here's my take, in diagram form:

The diagram is organized from basic (top) to advanced (bottom).
A solid arrow indicates a more or less definitive prerequisite. For instance, I consider numbers and sets a prerequisite for both groups and discrete math because working with sets is essential in both subjects, and because numbers and sets are a good place to learn to prove things. Also, many examples of groups arise from number-theoretic constructions. 
A dotted arrow indicates that one subject might be useful for learning another, although not essential. For instance, it would good to have experience in real analysis before learning topology, largely because topology requires so-called "mathematical maturity", which learning real analysis is likely to supply. Topology and geometry are linked to each other via a dotted line because general theorems from topology will apply geometric objects, and objects from geometry will supply good examples of topological spaces.  
The red bubbles are additional topics that came to mind. Perhaps you'll want to add some of these to your studies if you particularly like subjects that precede them. For instance, if you find you love basic abstract algebra, I'd recommend learning a little Galois theory. If you love topology, you might look into some algebraic topology-- study of the fundamental group. There are lots of other "elective"-type topics: if you come across something in your studies and find it interesting, consider changing your plans to include it. 
I haven't included calculus on the diagram. If you haven't learned that yet, you'll need to do so before vector calculus, differential equations, and complex analysis. Technically, you could learn real analysis without (or in place of) calculus, since most real analysis books rigorously prove results of calculus from scratch. However, depending on the analysis book, skipping a more typical introduction to calculus might be confusing: many texts seem to assume the reader has some sort of intuition for calculus.
I've omitted (15)-(17) because these are subjects with which I have less familiarity (that doesn't mean that I think you shouldn't study them, though! Just seek guidance elsewhere). For (16), if you have in mind a "methods of mathematical physics" approach, this could be studied soon after multivariable calculus.
I'd suggest that you pick a few subjects from the top of the diagram, and begin studying them. As you finish a subject, replace it with a successor on the diagram, or with a different topic near the top of the diagram. Over the next few years, you'll work "downward" through the diagram until you know all the material. 
General advice: do lots of exercises. If you're not sure about your answer and can't find a solution, post it on MSE for verification.
I hope that this is helpful, and isn't entirely overwhelming. Please let me know if I can clarify things.
Edit: to be more specific about where to start, vector calculus and linear algebra are the most canonical choices. I'd recommend studying them at the same time, if you are comfortable with this. If you prefer to choose one, vector calculus is probably best. See the comments on this post for further discussion.
A: This may be useful for a somewhat different perspective. My personal feeling is that it is (I would say) of foremost importance that you become comfortable and nurture a proficiency with proofs. 
The topics you mention, when studied in the depth I would think you want to explore all have a "proof component."
While this point could be argued, typically in most math curricula Real Analysis is the entry point to this amazing world. A good RA text (link to an excellent free notes by an outstanding teacher - Vaughan Jones)
https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtYXRoMTA0c3AyMDExfGd4OjJiNTJkM2M2ZWUzZGIwYWQ
will be self-contained and develop proof skills from a beginning level.
One cannot but appreciate the excellent answers given which specifically address your question. But, if I may suggest, pick a starting point with a rigorous, proof content. Once you have "completed" that course or text, you will have begun cultivating a degree of math skills - typically referred to as maturity. Then you can see where your interest lies for your next step.   
