Bach is the king of Baroque music - lots of running notes (not too many pauses in the music), generally the dynamics are a bit restricted compared to later styles (no huge/fast changes). Bach's music is probably the least overtly emotional music out of all of the choices you've listed, so at first you might find it a little dry compared to heart-on-your-sleeve romanticism or grand sweeping dramatism that you'd find from Liszt or Wagner - but there is plenty . Some good examples would be Chaconne in D minor (either the solo violin version or the piano version adapted by Busoni), Brandenburg Concertos, Mass in B-minor, solo partitas for various string instruments (violin, viola, cello) and keyboard, and lots and lots of preludes and fugues (48 for keyboard in the Well Tempered Clavier, numerous others for organ and other instruments). For most of the keyboard music that is now played on piano, remember that Bach played on harpsichords, which generally did not have the ability to have infinite shades of dynamics (usually there were loud and soft pedals, so that's why the music will have "terraced dynamics" where a phrase will be played loud or soft and then repeated with the opposite dynamic). Bach's style really didn't evolve or change that much over his lifetime.
Vivaldi fits in before Bach - early baroque/late renaissance. The Four Seasons is by far his most popular work and a good representation of his style. In general, small ensembles, mainly strings and perhaps a harpsichord, with a kind of verse/chorus aspect where an instrumental soloist (usually violin) takes the "verse" and the ensemble joins in on the chorus. Dynamics aren't all that diverse, and the music is fairly simple (with the exception of the solos).
Mozart is the most popular composer from the classical period, and you'll find a much lighter and more lively temperament in his music (generally, there are some very cool dark pieces Mozart wrote). I disagree with Lilly's statement that Mozart will generally write waltzes. While he does, I'd say more of his music is in regular 4/4 timeOne thing you'll notice in Mozart's music as compared to bach is the the melody really dominates - where in Bach's music there'll be several different melodies interacting with each other, in Mozart's music there is more often one main melody and then accompaniment (like the oom-pa-pa type accompaniment). I'd recommend any of his symphonies - The Jupiter is a good starting point - his later piano concertos, his string quartets, his later piano sonatas, and even a few operas if you're up to it (magic flute, marriage of figaro). With Mozart's keyboard music, he was using harpsichords at first and then the first prototypes of the piano, which had pretty limited dynamics and were pretty fragile instruments - so while there will be more dynamic change, it's still kept pretty polite. Mozart's style did evolve over his lifetime, but not too drastically - it generally stayed well within the classical idiom.
Beethoven is where it starts to get interesting.
Beethoven could be called the last of the classicals/first of the romantics. He really forms the bridge between the two genres - borrowing a lot of traits from the classical but adding a lot more passion and emotion into the music. The 3rd, 5th, 7th and 9th Symphonies are all outstanding, Violin Concerto #1, Piano Concerto #5, virtually all of his piano sonatas, and some of his string quartets should give you a good taste of the evolution from Mozart. Also, pianos were evolving and getting stronger and capable of great dynamic contrast, so that shows in beethoven's music as well. beethoven's style evolved quite a bit. His early period sounds a lot like Mozart and Haydn, his middle period starts to branch out a bit, and his late period really shows some very unique writing and very passionate and emotional music (the 9th symphony was written well after Beethoven was virtually entirely deaf).
Tchaikowsky is a good example of a romantic composer. The music is more free - more use of rubato (tempo isn't nearly as strict as Mozart, Bach or Beethoven). I'd recommend listening to his 1st piano concerto. Big, bold sweeping themes - lots of flash and flare, and tender melodies in slow sections.
Wagner is....interesting. I'd classify him as a late to post-romantic composer. Definitely wide-open heart-on-your-sleeves emotion. Wagner really opened up the use of the orchestra, with lush orchestrations using a much larger orchestra than Mozart or Beethoven would be accustomed to. Wagner tends to be pretty dark and brooding, and also you might notice that Wagner's large scale pieces tend to be pretty darn long
. I really haven't spent much time listening to Wagner's music - some of it can be really beautiful, but overall I find it to be too overdone.
I think I've written myself out...it's been a while since I've had a chance to talk about classical music...I kinda miss school!
I'll come back later to add in my own favorites from various styles.