As in most of the other arts besides painting, most notably music and poetry perhaps, there has been a radical break between the classical periods of these arts, say from the 17th-late 19th centuries, and their modern  period. The most significant aspect of this break has to do with the growing distance between the most advanced practitioners of these arts and their conceptual concerns and the wider, popular, public. Up until the second half of the 19th century there is a certain harmonious relation between the general public and the great art created within and for the public realm. But starting in th 1850's and perhaps a bit before there is a real break between the needs of the public and the interest of the artists. THese latter have been interested more and more in breaking the traditional rules of art, and in creating an art that is reflexive, that is, increasingly interested in its own modes of production, interrogating its own being being, rather than giving the public an image of its contemporary concerns. A growing alienation between public and arists have has been happening since that period (with the notable exception of film, the most recent of arts who remained for various reasons the only one inw hic the greatest works are potentially also the most popular and beloved. Due to this growing alienation between the public and the artists in all the major arts besides film, there has been developing an increasingly specialized vocabulary one NEEDS to learn in order to even start thinking and understanding what this modern art is about. General education is no longer enough to enable one to relate to the most advanced works of art. As such, the question of  learning to appreciate modern art is indeed a crucial one, since one indeed needs to lean a specialized way of thinking and relating to things. Perhaps the best way to start would be a close study of Kant's Third Critique, which might be the most significant treatise theorizing the main question raised in the thinkng about he arts in the last two centuries  (and one of the crucial dimensions of the complexity of modern art has to do precisely with the fact that, perhaps unlike previous art, it is essentially and intensely theoretical, that is, one cannot separate in relation to it theoretical issues "about" art, and so called simple appreciation of, and pleasure in, art). After Kant I would turn to a  fewmore key philosophical works  that were cruial to the development of a serious thinking about art, such as Nietzsche's the Birth of Tragedy, Heidegger's The Origin of the Work of art, Walter Benjamin's The Work of Art in the Age of Mechanical Reproduction, Merleu-Ponty's works on Cezanne, and Derrida's The Truth in Painting. I would then move on to read some famous art historians, Beginning perhaps with the works of Panowsky, Wolfflin, and Aby Warburg, and moving to contemporary critics such as Michael Fried, Rosalind Krauss, T.J Clark, and Georges Didi-Huberman. If you manage to cross this river, I think you will be on your way to understand the crucial issues that are at stake in the great works of the last hundred and fifty years.

An extremely interesting theory Dave. I never heard it before - is it yours or a well known one?

Though I find the theory very interesting, I completely disagree with your suggestions. I don't think you need to know theory at all to appreciate modern art. What you might have to accept in order to enjoy painting at least, is that a painting of a chair is not there to show you "the true chair" but something much more general. In this sense modern art alters our view not only of it but of the entire of corpus of art before it, or rather as you say 17-19th centuries. 

I would tend to say that modern art is a different kind of language which you need to live with for a while before understanding, like any language. There is obviously something we see as easier in romantic art and more approachable, but we should remember that during those times, anything modern sounded like nonsense to them too. That is the essence of change.
To cut it short, as everything here is clear, I would just say that if you want to understand modern art, live with it a bit. Listen to a lot of modern composers - Schöenberg, Varese, Hindemith, Janacek Bartok, Boulez to name just a quick few. Notice to choose very good interpretations of them, as most interpreters don't understand them either so how are they going to help you understand them. (I appended a few possible discs, just on the top of my mind, discs I liked.) There are of course many many more. Just listen a lot to them, even at the beginning you don't understand their language. It might help to start with their close roots like Debussy, Satie, janacek; again, just to name a few. The name I threw are more just what is coming to my mind this minute and I'm sure I'm missing many extremely important ones. But this is not important.

To reiterate, always in music, if you want to understand or appreciate anything, choose your interpreters carefully. They are just as important as the composer and piece you choose to listen to. At least if you want to get something out of listening beside a nice pass-time. 

Similarly for painting. Look well at Miro, Pollock, Richter, Rothko (to name a tiny few) and maybe their origins of Cezanne, Matisse, and Gustave Moreau. Just allow yourself to look seriously for a while, and you will start to understand.

It is no different than starting to listen to classical music nowadays, or look at classical art in general - this too takes practice and a slow entrance to the field.

Music Discussed

I was very excited to attend a recent gallery opening for a modern art show with a highly respected art curator friend of mine.  Finally, I could find out the hidden meaning behind these objects!  Before I could ask her what they all meant, we overheard a conversation that went like: "what do you think the artist meant by this?"  "I don't know... I don't get it."  My friend started laughing and said "why do people think there is a hidden meaning?  It just means whatever you want it to mean.  How does it make you feel?"