How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

There are several things you could study.

Topology is definitely a must, especially when you start talking about bifurcations, and it sounds like you have a chance to take that. That could be a great first start, especially as an undergrad.

In the long view, since you're coming from a physics background, you might want to learn Hamiltonian mechanics and start looking at things like integrable systems, perturbation theory, and the stability of the solar system -- which is something you'll probably do at some point anyways. (Besides, this is historically how the subject got started.) You may get some of that in an advanced undergraduate class on classical mechanics (they should at least touch on Lagrangian mechanics), but probably not much. An excellent segue is V.I. Arnold's Mathematical Methods of Classical Mechanics. Another very good book, and heavy on the math, is Goldstein's Classical Mechanics. Both of these are considered graduate-level texts but you sound like you can handle them.

About the math involved: if you haven't had a class on real analysis yet, you might want to think about that. If you have, measure theory might be the next logical step. Differential geometry could also help, although that might be a bit ambitious, and besides, Arnold develops it as he goes. If you don't get this as an undergrad, it may not be a big deal, but you should certainly think about all of these things in grad school.

Some of the big guns you might aim to understand are the KAM theorem and things like the Poincare-Bendixon theorem. If I remember, Strogatz gives a great discussion of the latter.

As soon as you start talking about strange attractors, the possibility of fractals comes in, and then topology and measure theory will be helpful, so that you can talk about things like the fractal dimension of your basin of attraction. One of my favorite examples of this is the magnetic pendulum which has wonderfully strange properties.

If it's offered, you could consider taking a course on Statistical Mechanics, where you would hopefully get introduced to the ergodic hypothesis. In this context, you might want to look at things like the Poincare recurrence theorem, applied to, say, a gas in a box. This could give you a feeling for how measure-theoretic questions and ergodicity can enter into dynamical systems. You might also look at Arnold's Ergodic Problems of Classical Mechanics.

In general, Scholarpedia is an excellent source on all of these topics: it's like wikipedia on math-crack. They have a whole section devoted to dynamical systems, with articles written by people who made significant contributions to the subject.

Here are some items for you to think about and review.

Nonlinear Dynamics:

Undergraduate courses in Calculus and / or Differential Equations, Dynamics, Linear Algebra, Mechanical Vibration

Texts:

Steven Strogatz, Nonlinear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering, Westview Press, 1st edition (Edit: as of 2014 there is now a 2nd edition containing additional applications.)

Holger Kantz and Thomas Schreiber, Nonlinear Time Series Analysis, Second Edition, Cambridge University Press.

An Introduction to Dynamical Systems, D. K. Arrowsmith, C. M. Place

Introduction to Applied Nonlinear Dynamical Systems and Chaos, Stephen Wiggins

Scratchpad Wiki

Suggested Optional Texts:

(1) Julien C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, 2003,

(2) Lawrence N. Virgin, Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, Cambridge University Press, Cambridge, UK, 2000, and

(3) Chaos in Ecology: Experimental Nonlinear Dynamics, Academic Press, Elsevier Science, 2003.

Review Nonlinear Dynamical Systems and Chaos Review MediaWiki Nonlinear Dynamical Systems and Chaos. See the reading materials listed to give you an idea of the prerequisites for you to consider.

Fluid Dynamics Classes in Dynamics, Calculus III, Differential equations

Reference books (Math oriented and there are many others)

Theoretical hydrodynamics by L.M. Milne-Thomson

An introduction to theoretical fluid mechanics by S. Childress

Fluid Mechanics by Kundu and Cohen

Fundamental Mechanics of Fluids by I. G. Currie

A mathematical introduction to fluid mechanics by Chorin and Marsden

Fluid dynamics for physicists by T. E. Faber

Physical fluid dynamics by D. J. Tritton

Regards