Warning: For the purposes of this overview, the first person assumes a very arrogant and belligerent personality and extreme viewpoints. This shouldn't be a problem: such an approach helps crystallize opinions and wash away insignificant details and compromises, leaving the naked truth for all to see. :)
Let me consider this most pressing topic first. The identity of the "best composer of all time" is of course an important point of view that everybody must have. If you don't know who the best composer is, you can't claim that you consciously listen to classical music, since the only purpose of such listening is finding out whose music is best.
The baroque style period gives historically the first precedents of music still worth anyone's attention today. Notably, you can still buy records of Bach (was it C.P.E. or P.D.Q. Bach?) and of Handel's Messiah in the cheap "best hits" section, so it cannot be so bad. Vivaldi's synthesizer software liked to get stuck in long loops repeating the same short triplets or quadruplets a thousand times over. This was an easy way to get several hundred concertos out. So I recommend Vivaldi if you need a sonic wallpaper. Of those guys back then, Bach is really the best. Listen to his... his... er... whatever he used to write those days. It was all played automatically from his keyboard's cache memory -- that's his own words.
Classicism essentially means Haydn, Mozart, Beethoven. It was an era of ear candy; people realized that they can make music that sounds nice, tastes nice, smells nice, looks nice, isn't too difficult to play (well, mostly), is popular with local authorities---and yet completely lacking content beyond what can be appreciated by a 5-year-old. Well, what did you expect of that 5-year-old composer who never grew up. (Well, if he did grow up, it was when he started listening to Bach and wrote that shameless plagiarism, the "Requiem" which went unnoticed because those days nobody had ever listened to Bach.) The number of Haydn-and-Mozart like composers must have been truly stunning at the time. But the great three Viennese classics made sure they held rights to the genre: Haydn created an infinite number of infinitely short symphonies, Mozart wrote innumerable concertos for any instrument, and Beethoven authored a similarly unbounded number of three-part piano sonatas.
After the stifling era of writing the same sequences of chords for almost a hundred years, there came the romantics. The preferred one here is Chopin, a visionary who first realized that the piano had pedals you could step on, and that keys make different sounds when pressed quickly or slowly. Duh... but what can you expect from the 19th century people anyway. The ones who came after Chopin were worse. Most of them were an odd bunch: Schubert broke his wrist tendons because he thought longer fingers would make him a better musician, and Schumann just couldn't get over that. Their imagination ran wild around the already somewhat beaten path of "roses-mimoses" and "flood-blood" rhymes. Liszt, being somewhat more cool-headed, started out as a "better Chopin than Chopin" piano player but somehow also managed to persuade everyone to think he was a composer as well. He seemed not to care what he composed as long as it had twice as many notes per second as anybody else was able to count. Nice try: today's piano students have to play it all anyway.
People born around 1870 started writing rather extreme music. I especially note three composers born almost in the same year: Rachmaninov (1873-1944), Reger (1873-1916) and Scriabin (1872-1915). It's interesting that the last two of them died around the same age. The first two wrote analogous things, Rachmaninov for the piano and Reger for the organ: pieces that were trying to crush too many notes at once in an attempt to render some overly important things into music. Rachmaninov was a good pianist while Reger wasn't a good organist, and it showed. However, I am sure Rachmaninov would not have enjoyed the present worldwide fame had he died at the same age as the other two guys instead of moving to the US and essentially becoming a concert pianist and conductor instead of writing any new music. Scriabin, unlike his college-mate Rachmaninov, underwent a radical change of compositional style around 1905, and another one around 1910; had Scriabin been less insane or had he lived longer, the world, or rather, the whole Universe, would have been changed forever and irrevocably (or so he believed).
Then there were Debussy and Ravel, the "impressionists". Debussy primarily concentrated on composing long and fascinating French titles to his very short piano pieces, Ravel, on the other hand, admitted to not being able to "compose any new notes" but nevertheless had a big success after doing some exercises in orchestration.
Thereafter, music could never remain as it was before. Well, it did for some diehard old-timers like Tschaikovsky (a big producer of chocolate au lait colored candy-music) but they don't count.
Ten years later, all of this was already much too orthodox. By 1950 it was clear that the 20th century stands firmly on its own with its own music. We learned not to wince at dissonances or when all keys are pressed at once; we knew things could get much nastier. Hindemith, the great theoretician, knew it all. Enter everyone: the crushed, neurotic Shostakovich who alternated between writing odes to Stalin and dissident music, and Prokofiev, now officially the best Soviet composer, still struggling to create art acceptable to the regime; enter Dupré and the whole school of modern ear-shattering French organ music, Messiaen sunk deep in his non-retrogradable nirvana.
Add your own favorites here and don't forget that Dupré could play it all from memory without making mistakes.
"Yuck. Bleah... Yuck. Bleah... Shlopppp... Yuck. Bleah..." The piece lasts for about 4 and a half minutes... Finished. The musician stands up. "Thank you. Thank you, I appreciate it... Please meet the author..." The musician bows out and the people go home still trying to get it: they had just listened to a few minutes of strange and perhaps somewhat impolite noises under title of a "Prelude #2 for a prepared washing-up bucket and a wet mop." Of course, it's all music, how can it be anything else but music. It's just that... hm... we need to think a bit more about the interpretation.
Today's music making seems to be very different. What they call "music" these days is mostly singing accompanied by loud, synthetic-voice melodies and chords; all is played in strict tempo and largely without dynamic expression. For some reason it is always the same genre piece: homophonic dance rhondo, alla breve, with obligato drums, for voice, guitar, keyboard and drums. And somehow they still subdivide this piece into a zillion different "styles". What's most puzzling is that, judging by the category labels in CD shops, the whole variety of the more traditional music is dumped into one "style": the "classical style"! As if all classical music were yet another way to sing to the all-pervasive dance rhondo, alla breve.
Nowadays, I'm told, what's most important is the words being sung (called the "lyrics"). Let's see... so what exactly is it they are singing? "Oh, yeah, baby... ah-wanna-have-yah-body... forevah... togethah... duh, duh, duh-duh..." Repeat.