That sure sounds like the text I use for my graduate series (though I only finished the first two quarters and haven't taken the qualifying exam yet). I was going to take complex analysis instead of real analysis (I heard the exam was easier), and I finished pretty much everything else. I've had a year of graduate differential geometry (differentiable manifolds, riemannian geometry, integration on manifolds), Lie Groups (differentiable manifolds, Lie groups, Lie Algebras), mathematical method in physics, and some odd classes on quantum field theory and relativity to kill time. Most of my research has been in differential topology which I got interested in when I was working at the NFS Geometry Center a number of years ago. One of my first papers was on tight immersions of manifolds (both polyhedral and smooth) and I was lucky enough to work with a few Fields Medalists who helped catch my mistakes during my work.
So has any area peaked your interest yet? My interest in theoretical physics pushed me into mathematics (though the fact the our department is mainly experimentalist didn't help stop me from changing majors). All the best physics was being done in our math department, and the physics department was thirty years behind in their math (I learn relativity coordinate free while the physics department was still using tensors which doesn't show you the global characteristics, but is good for calculating paths).