Taking the theory and musicianship classes at the local University or Junior College is a good place to start.
I've always felt Classical to be musician's music (perhaps a kiss of death in this age) and so a working, firm knowledge of theory and its application are indispensable in listening to Classical. Style from a Historical point of view, is a great asset, as well—are they playing Baroque like Baroque? Are they avoiding playing Mozart’s fortes, like Beethoven or Wagner’s?
It has also has the advantage of edification, for me at least--I cannot count the number of times I’ve been overtaken by a sense of awe, inspired and invigorated by the pure genius of some composers; Mozart's use of the Neapolitan in the second movement of his 23rd Piano concerto, as well as the subtle hints at "mi"; Beethoven’s use of cross relation of f-natural and f-# with a tonicization in the 18th measure of his violin sonata no. 7; Schubert’s text painting in Der Erlkönig. These are all subtleties I find fascinating because they all appear to hint, or imply an experience the composer is attempting to transmit.
Hope that helps, it’s some of what I’ve learned thus far.