Here's a couple that haven't been mentioned yet:
Music for the Royal Fireworks - G F Handel
Variations on a Theme By Paganini - Rachmaninoff
Folk Song Suite - Ralph Vaughn Williams
Good luck on your discovery of Classical music!!
It is impossible for me to list all the great classical pieces I know as I have become steep in it longer than most members have been on the planet. I also have performed it professionally. I am going to take the very popular Composer Ludwig van Beethoven only and name his most popular pieces only:
Symphonies: 3, 5, 6, 7, 9
Piano Concerto: 5
Violin Concerto: he only wrote one.
Piano Pieces (solo): Fuer Elise, "Appasionata," "Moonlight"
Overtures: Egmont, Leonore, Fidelio, etc.
These specific pieces are so popular, they sell tickets. It is also important to get the best performers of each piece, as there is a lot of good but less than superlative recordings out there. The more you get to know the pieces the more you will be able to tell.
Also, live performances are what the music was written for, so they sound best live. If you want more, I can help you by message.
I grew up in a fairly classical household. Dad was into operas and symphonic music as well as lighter fare like Gilbert & Sullivan and the Strauss family waltz music. So, I kind of picked it up from him and talked about that in my part of the eulogy at his funeral a couple years back. My favorites are the Beethoven symphonies and piano sonatas. In terms of opera, Wagner and Puccini are my favorites though I'm fond of Verdi, too (saw Falstaff in the spring at our local university music faculty). I played clarinet in high school and still pick it up from time to time but I've never been very good at it and if I was going to pursue it more seriously now, I might go for jazz rather than classical.
Vivaldi The Four Seasons - Spring
I love this piece