It's great that you're in this mindset! I should warn you that you will probably always feel like you're 'behind' in concert-music appreciation and that's because there's just so much of it. Even people who really know what they're talking about have to have their own niche, whether it be music of a specific time period, for a specific instrument or ensemble, or of a particular composer. Being truly 'well-rounded' in the world of western concert music is sort of an unattainable goal that. I hope that doesn't sound discouraging, because to me that's one of the most exciting things; there's such a variety of ideas and means that there's always something new to learn.
So, anyways, I am guessing that the only reason you don't like a lot of 20th/21st century music is that a lot of it is obscure and you haven't heard all of it. I admit that 12-tone and other forms of serialist music are often more interesting on an intellectual level than on a musical or aesthetic one (in other words, they're not easy listening), and that I don't even like playing serial music that much. The great thing is, not all atonal music is necessarily like that. Atonal doesn't necessarily imply a complete lack of triadic harmony, just the neglect of the typical triadic musical syntax that we call functional harmony (things like ii-V-I). Much of Stravinsky's music is arguably atonal, even outside of his experimental 12-tone works. The point here is that things can sound tonal while not necessarily following all the rules of tonality. The piece that I'm really thinking of right now is the first Piano Sonata by Carl Vine, which is very much atonal, but never at the expense of recognizable thematic development or melodic interest. There are even sections in the first movement where he comes to a fairly tonal climax in B major. So, while this music is atonal, it is still quite accessible to someone aquainted with Stravinsky and the occasional sprinkling of Schoenberg, and it's definitely not serialist. Vine's other two piano sonatas are quite similar, as are his symphonies. His Piano Concerto is even more based in triadic harmony, though it is still far from 'tonal' in a traditional sense.
Next on my list of composers I wish people would listen to more are Ligeti and Lutoslawski. These are two giants of eastern european classical music who died as recently as 2006 and 1994, respectively. I actually just listened to a really interesting, if poorly presented, talk by a Yale musicologist about the use of triadic harmony in Ligeti's music, which is again 'atonal' in a strict sense. His music often creates triads as a result of horizontal voice movement rather than, as in Vine's, using them as a coloristic tool. He has some very interesting works, which often draw on folk song in a sort of Bartok-ish fashion. I like his Wind Bagatelles, Piano Etudes, Horn Trio, and String Quartets.
Lutoslawski is a bit different, definitely a bit easier to grasp in a developmental sense. I was introduced to his music by way of the Paganini Variations for 2 Pianos, which remains one of my favorite collaborative pieces. By virtue of it's subject matter, it is necessarily more 'tonal' than his other works, but that doesn't stop him from transforming that well-known theme into a complete atonal carnaval by the last variation. I am also a huge fan of his first symphony, his piece 'symphonic movement', and his piano concerto.
One more; Samuel Barber. He's often classified as neo-romantic because he was so fond of lush harmony and gushing melodies. However, he was also fond of using serialist techniques and atonal syntax. His great achievement is in combining the acessibility of a tonal piece with some of the techniques of atonal composers. In his very romantic 'Nocturne', for example, he uses a tone row (!) as the basis of the melody and then develops it in a quasi-serialist manner, while still managine to build an almost sonata-form piece with a very accessible form. His violin concerto is one of my all-time favorite pieces, definitely moretonal than atonal. The first symphony is also a great piece (one of my favorite pieces to listen to live!), and surprisingly accessible while still being very intellectual. Knoxville; Summer of 1915 is a concert favorite, but by far my favorite work of his is the Piano Concerto Op. 38, not just because it is the everything-and-the-kitchen-sink ofconcertos, but because of the ingenious way he synthesizes the opening atonal melody with the second theme and other musical materials, while remaining highly accessible.
I back everything the previous answerer said about Messiaen! What a great composer, orchestrater and notater of birdsong! He definitely has the most unique sound world and tonal language of any composer I've ever heard. (As for what someone else said about Prokofiev: he's atonal? No way! Tons of chromatic and extended harmony maybe, but definitely tonal).