Anything by Mozart- I’m surprised he hasn’t been mentioned more. My favorite Mozart pieces are Symphony No. 29 and String Quartet No. 19 in C Major (nicknamed the Dissonance Quartet).
Classical music is so diverse, it’s hard to know what will speak to an individual, but here are some of my favorites:
Bach’s 2nd Brandenburg Concerto - so full of joy and confidence and yet still so mathematically structured. If you love math, you’ll love Bach.
Mozart’s Clarinet Quintet in A Major - this is achingly beautiful and the precision of the interplay between the instruments is amazing.
Dvorak’s New World Symphony - This is the gateway to classical musical for many people. It’s eminently listenable to the rookie because the music is so evocative you can easily see the pictures Dvorak is painting. The little donkey’s theme is sure to bring a smile to your face.
The Chopin Nocturnes for solo piano - Most of these are fairly short, so good for a beginner. And I guarantee they will move you , particularly the E Flat nocturne. It’s astonishing how much emotion Chopin could put into these pieces with such economy of notes.
You have two choices for a piece entitled “Scheherazade” in the standard repertoire, none by Prokofiev:
Rimsky-Korsakov, Scheherazade.
Ravel, Shéhérazade. Note that this one is either a song-cycle based on poetry about Scheherazade, not the original stories, or an unknown orchestral work. Odd that there are orchestra pieces by Ravel that are unknown! But so it is.
ETA: As for the OP, I couldn’t possibly choose. I’d be hard pressed to even choose a favorite composer from each major time period.
Yay, Ligeti!
But no Lutosławski?
Rachmaninoff: Piano Concerto No. 2 in C Minor, Op. 18 - the first movement in particular. It was the first classical piece I really enjoyed that wasn’t one that is instantly recognizable by laypeople. It still remains my favorite.
If you’re interested in checking out the Romantics:
Gustav Holst: Brook Green Suite; Saint Paul’s Suite
Maurice Duruflé:* Quatre Motets sur des Thèmes Gregoriennes*
Modest Mussorgsky:* Pictures at an Exhibition*
Gerald Finzi: Five Bagatelles; Three Soliloques from “Love’s Labor’s Lost;” Severn Rhapsody
David Diamond: Rounds for String Orchestra
This is what I would call Modern Classical, or most people would say Classical Lite. But it’s beautiful, emotionally affecting music.
All right, I’ve give this a shot. Here’s a shortish list of pieces that have been massively important to me (as you can see, I couldn’t even stick with just one piece by one composer in some cases!):
Josquin: Missa pange lingua
J.S. Bach: Passacaglia in C Minor, BWV 582
Mozart: Symphony No. 41 in C Major, K. 551
Beethoven: Symphony No. 4 in B-flat Major, Op. 60 or Symphony No. 6 in F Major, Op. 68 “Pastorale”
Schubert: String Quartet in G Major, D. 887
Verdi: Falstaff
Bruckner: Symphony No. 8 in C Minor (really hard to just pick just one Bruckner Symphony)
Strauss: Also sprach Zarathustra, Op. 30 or Ein Heldenleben, Op. 40
Mahler: Symphony No. 3 in D Minor, or Symphony No. 6 in A Minor, or Symphony No. 9
Nielsen: Symphony No. 4, Op. 29 “The Inextinguishable”
Sibelius: Symphony No. 3 in C Major, Op. 52, or Symphony No. 4 in A Minor, Op. 63
Stravinsky: Le Sacre du printemps, Symphony of Psalms, or Requiem Canticles
Bartók: String Quartet No. 4 or Concerto for Orchestra
Debussy: La Mer
Ravel: String Quartet in F
Schoenberg: Variations for Orchestra, Op. 31 or Chamber Symphony No. 1, Op. 9
Berg: Wozzeck or Lyric Suite
Webern: Five Pieces for String Quartet, Op. 5 or Symphony, Op. 21 or Cantata No. 1, Op. 29
Varèse: Arcana
Holst: The Planets
Prokofiev: Symphony No. 5
Shostakovich: Symphony No. 13 “Babi Yar”
Boulez: Notations I-IV for orchestra
Lutosławski: Symphony No. 3
Ligeti: Lontano, Chamber Concerto, Horn Trio, or Lux Aeterna
Carter: Variations for Orchestra
Adams: Harmonielehre or Guide to Strange Places
Yeah, a barely manageable list. Sorry, folks. I actually hit submit to prevent myself from adding more to it.
My favorite piece ever is /Adagio in G minor, nominally attributed to Tomaso Albinoni, but actually composed by Remo Giazotto. Nevertheless, it is quite the haunting piece. It is often used in films - it was used effectively in the original Rollerball, and is the entirety of the soundtrack for Orson Wells’ Kafka’s The Trial.
My second would be the Waltz from Sleeping Beauty, by Tchaikovsky. I always want to sweep around the ballroom when I hear that one.
I’d add Beethoven’s Moonlight Sonata, Strauss’ Blue Danube, the already mentioned Rimsky-Korsakov Scheherazade, and the dances from The Nutcracker Suite.
Dvorak: Serenade for Strings
Debussy: Clair de Lune
Paganini: 24 Caprices
Rodrigo: Concierto de Aranjuez
Great topic here - I offer:
Sibelius Violin Concerto
Vaughn Williams (early and late)
Norfolk Rhapsody, In the Fen Country, The Lark Ascending, Serenade to Music and Symphony No. 7
Henryk Gorecki Symphony No. 3
Here are a few I haven’t seen mentioned yet:
Some of Handel’s Organ Concertos showed me that organ music could actually be fun. Op. 4 #4 is a catchy one. For a more modern take on the organ concerto, try Poulenc’s.
String quartets: Haydn wrote a lot of them that are clear and easy to appreciate. Try “The Lark” (Op. 66 #5). Then for something a bit more Romantic, try Dvorak’s “American” string quartet (#12).
Piano Trios: Definitely start with Beethoven’s “Archduke” trio. It’s what turned me on to this particular form.
Mahler’s 5th symphony is long, intense, and emotionally satisfying.
There are a lot more pieces I could mention, but that could take all day…
I don’t have much to add here, but I want to second this:
I’m surprised it’s mentioned only once above.
Rachmaninoff: Piano Concertos #2 and #3; Symphony #2.
Chopin: just about everything he wrote.
Tchaikovsky: All the Symphonies; all the ballets; Piano Concerto #1; Violin Concerto.
Brahms: Symphonies, especially #3; Violin Concerto; Piano Concertos.
Verdi: Rigoletto.
Bizet: Carmen.
Mozart: Symphonies #39, #40, #41; Clarinet Concerto; Clarinet Quintet.
Beethoven: Symphonies #3, 5, 7, 9. Piano Concertos.
Kabalevsky: The Comedians.
Saint-Saëns: Piano Concerto #2; Violin Concerto #3; Carnival of the Animals.
Lehar: The Merry Widow.
Bach: Brandenburg Concertos.
A couple well-known Mozart piano pieces:
Rondo alla Turca
Piano Sonata in C Major, KV545
There you go 
:
Lutoslawski: Symphony n°3 + Piano concerto + Trois Poèmes d’Henri Michaux.
Which forces me to add:
Dutilleux: Tout un Monde Lointain (for cello and orchestra) + Ainsi la Nuit (for string quartet) + Symphony n°1
Tchaikovsky : Violin Concerto in D major op.35
It’s rather obscure, but Vivaldi’s Concerto for Diverse Instruments, in C.
I’m surprised I’m the first person to mention Mendelssohn. The first movement of the Italian Symphony is among the most accessible and enjoyable classical pieces for newbies. And if you know “A Midsummer Night’s Dream,” the overture he wrote is wonderful, too.
Beethoven’s 9th Symphony is arguably the best music ever written. And Beethoven was almost, if not completely, deaf at the time he wrote it. It is from the Romantic period, so technically not “classical”, which would be from the end of Baroque (J.S. Bach and such) to the start of Romantic (Beethoven and his crowd). Classical includes the Haydn boys, Mozart, Saliere and those gang bangers.
[quote=“furryman, post:11, topic:715193”]
The William Tell Overture
[/QUOTE]It is the sign of cultural maturity when a grown man can hear the William Tell Overture and not think of The Lone Ranger.