I recently came across noncommutative geometry and found it rather interesting. I should mention that I'm a graduate student considering options for my research and if I were to name an area which I'm interested in, then it would be functional analysis including operator algebras etc., and that was how I got to know about noncommutative geometry. From what I've been told, noncommutative geometry is a very broad area so I would think that one can go in many different directions after entering the field. What I would like to find out is what some of these directions are. Also, if I am to go into this field, my motivation will probably be functional analytic so I will be particularly interested to know if there is an approach that suits me.
1)You look into the book "Noncommutative Dynamics and E0 semigroups" by William Arveson 2) There is an approach to attack multivariate operator theory through algebraic geometry.You may look to "Operator Theory and Complex Geometry" by douglas for an introduction 3)You may look into brown,douglas,fillimore's paper in essential normality and K-Homology which was a precursor to connes approach 4)or look into connes's book
Connes' book is pretty tough to get through as a beginner. I would suggest as an alternative the book Elements of Noncommutative Geometry by Gracia-Bondia, Varilly, and Figueroa. Or for a more concise, but less thorough, introduction, I like Varilly's book An Introduction to Noncommutative Geometry, which is also a little more recent.