I've been pondering what music of Mozart and Beethoven to recommend. These two really are the pinnacle of Classical Music. Mozart defines "prodigy," and when you think of the Symphony--everything either leads up to Beethoven symphonies, or follows them--he defined "symphony," as well as many other genres.
Mozart's father was a well-known musician, and so he learned to play the Clavier (or whatever piano forebear it was at the time) at a very young age. He composed his first piece of music at age six. He and his sister, "Nannerl," performed all over Europe.
Unfortunately, a lot of my knowledge about Mozart is overshadowed by "Amadeus," the movie/play about how Salieri killed him out of jealousy. It portrays Mozart as silly and horny, and utterly ridiculous, which is unfair. There are lots of theories about him, though, and a recent one is that he was on the Autism spectrum, which could explain his genius in some areas, and then lack of common sense/social grace. Whatever the truth, Mozart's popularity as a child prodigy could not sustain him as an adult. So, in spite of his genius, he couldn't hold jobs for very long, and he did die very young, and was buried in a pauper's grave.
It's best to focus on his genius. When you look at his rough drafts, you see it all very neatly written out, with very few corrections. The music would be fully formed in his mind before he wrote it down, and it was perfect.
When you compare Mozart's with Beethoven's scores, you see a huge difference in the thought process. Beethoven's scores are unreadable. He would work out every single note, measure, phrase over and over again until he got it right. Beethoven's music tends to be more focused on the development of a motif, while Mozart's is more melodic.
Music by Mozart: Eine Kleine Nachtmusik--very overdone, however, I can actually remember humming this when I was walking to school in second grade. For kids, they're too young to have heard anything too many times, and it is a great piece.
He also wrote a theme and variations on Twinkle Twinkle Little Star for piano--they might enjoy hearing what a composer does with a familiar tune.
I used to have a pirated copy of Mozart Overtures performed by Neville Mariner and St. Martin-in-the-Fields, and I highly recommend it still, though I got rid of my burned tapes a while ago. The overtures capture Mozart's sunny personality, and his flair for drama.
Mozart's last three symphonies are three of the best works ever written--nos. 39, 40, and 41 ("Jupiter"). Also, I love his "Haffner" symphony--no. 35.
Beethoven was a talented youth, but had a very unhappy childhood. His father, wanting Beethoven to be the next Mozart, would hit his hands with a stick if he made mistakes while he was practicing. Beethoven toured Europe as well, but his father lied about his age, saying he was two years younger than he really was, trying to cash in on his son. When Beethoven was 16, he performed some of his own compositions for Mozart, and Mozart said to his wife, "Remember that boy's name. One day you will hear it often," or words to that effect.
Beethoven's fame grew more slowly than Mozart's, but in the end, he was terribly famous, and he had a grand funeral. People stole locks of Beethoven's hair--he was quite a celebrity, though I don't think he was ever a happy man.
He was famous for his temper, which can be explained by so many things...There's his abusive childhood, his numerous romantic failures (he never married), and of course, his loss of hearing, beginning in his twenties, until he was profoundly deaf in the final years of his life.
It's the fact that he was deaf when he wrote some of his greatest music that is so inspirational. He had the legs taken off of his piano to set it directly on the floor, so that he could feel the vibrations of it when he played. (Also, it is told that he broke several pianos, even before he was deaf, because he played so hard. One of his sonatas, the "Hammerklavier," was written for a new kind of piano, one that was of a stronger build. It's not the piano as we know it today, but was a step closer.) His knowledge of music theory was tremendous, too, for he was able to construct his compositions based on what he knew, rather than what he could hear.
A famous story is of the premier of his 9th Symphony--his final symphony, which he conducted, in spite of being deaf. (I believe there was another conductor on the stage, whom the musicians followed, while Beethoven waved his hands around to what he was imagining.) When the piece was over, he was probably still conducting, when someone turned him around to face the audience, which was standing and cheering for his great achievement.
Beethoven is often called "The Giant Astride Two Centuries," because not only was he born in the 1700s, and died in the 1800s, he went from being a "Classical" composer, to being a "Romantic" composer. All composers after him had to deal with his greatness--they each had to figure out how to follow his precedent--and those are stories you learn when you read about later composers over and over again.
So, early Beethoven works have a different sound than later ones. You can hear the progression when you compare music from the same genre: Piano Sonatas, Piano Trios, String Quartets, and so on.
I'm stalling, because I don't know what to recommend. Well, off course I do--Beethoven's 5th. Probably the most famous music ever written--and it cannot be over-played. So many recordings, too many to single one out as being "definitive."