I guess it’s neoclassical, but Cantus for Benjamin Britten by Arvo Part. Great swellings of strings against a church bell. It’s an astounding piece. The album that it’s from, Tabula Rasa, is profoundly moving.
Mendelssohn’s Violin Concerto in E minor, 1st movement (although it’s not that slow)
2nd movement of Beethoven’s Piano concerto #4 in G Maj Op. 58:
ii. andante con moto
As far as your car goes, MP3 players are pretty affordable now and can jack directly into most modern car stereos, as well as be powered by your car’s battery (most modern smartphones also have this capability, but with less storage capacity; they’re more expensive too, but if you already have a smartphone anyways…). Failing that, most modern car stereos can play MP3-format discs, which can hold a fairly large amount of music (several hours worth per disc).
The slow movement of Saint-Saens Symphony #3 (Organ Symphony). Serene mostly, but introspective.
Leos Janacek’s “In the mists”
Leos Janacek’s “Sonata 1.x.1905”
Domenico Scarlatti’s sonata L.108 / K.213
While you’re listening to Barber, also check out the second movement of his Violin Concerto, Op. 14.
Beaten to it!
Nitpick: Otto Klemperer. (a.k.a. Maestro Klink.)
I also really like Introduction and Allegro by Elgar. Oh my god is that an awesome piece!
The 2nd movement is closer to what he’s looking for.
And Tchaikovsky’s “The Seasons: June” (except for a brief boisterous section in the middle.
Try Shostakovich Str Qrts – all of them. There is something of that experience in any composer, ceteris paribus.
Mahler?
Vaughan-Williams?
And so forth. Listen to it all. And learn to know what feelings are.
Welcome to the world of music.
OK, I see on specifics. I like - YouTube Bach’s long fugue from the e Partita for your mood.
But also the Brahms Op. 118 no. 6 – the “Dies Irae” based intermezzo (which is, I guess, supposed to be a lighthearted piece, and this one is, but one old friend said when I was playing it again, “How can you stand to play so much beauty?” (srsly it was like a weird moment, since I always liked it and just thought it was normal). Look up Luba Edlina’s performance, I guess. Gould is OK, but why not get some thunderous octaves under your belt first. 
Special mention for the last one, No. 15. Six movements…all marked “Adagio.”
Debussy, nocturne for orchestra “Nuages” (“clouds”) Debussy: Nocturnes - No. 1. Nuages, Conductor: W. Sawallisch - YouTube
Debussy, prelude “Footsteps in the Snow” Debussy: Book I~VI. Footsteps in the Snow - YouTube
Ravel, Mother Goose suite, sections 1 and 2 RAVEL - MOTHER GOOSE SUITE : 1. Pavane of the Sleeping Beauty - YouTube
Ravel, Valses Nobles et Sentimentales, sections Assez lent, Presque lent, and Epilogue: lent Weissenberg - Ravel Valses nobles et sentimentales - YouTube
Holst: The Planets - Neptune - YouTube
Chopin prelude in B minor F. Chopin - Prelude No.6 in B Minor, Op.28 - Evgeny Kissin - YouTube
Bach, sarabande from Orchestral Suite in B minor Bach - Bwv1067 Orchestral Suite - 03 - Sarabande - YouTube
Bach, sarabande from unaccompanied cello suite #2 - YouTube
Bach, Largo from Concerto for violin in G minor, BWV 1056r Bach Violin Concerto In G Minor, BWV 1056 - Largo - YouTube
Bach, Adagio from Concerto for violin in E, BWV 1042 J. S. Bach: Violin Concerto in E major - 2nd movement - YouTube
Bach, sarabande in D minor for unaccompanied violin, BWV 1004 Anne Sophie Mutter - J. S. Bach Sarabande - YouTube
Bach Suite 5 for unaccompanied cello IV. Sarabande - YouTube
Youtube breaks this up as 1/2 for 95% of its listings and that part didn’t hit me like I thought it would. Faith in poster, I dug further to find one of the few 2/2 listed.
So. Worth. It. Thank you!
It’s written in 2 movements- “The Presentiment” and “The Death” - and tells the story of a student protester who was killed. The second movement is chilling. My favorite performer of all things Janacek is the pianist Leif Ove Andsnes.
Elgar - “Nimrod” from Enigma Variations
The Adagio from Khatchaturian’s Gayane is about as sad is it comes. One of my favorite pieces, and may be familiar because it’s used quite hauntingly in Kubrick’s 2001.
Of course, in the entire suite, Sabre Dance comes in a little later and picks up the pace.
This is the first one I thought of. One of my absolute favorites.
Amen. This piece is stunningly beautiful transcendental impressionism. Given a choice, I’d go with the Ormandy/Philadelphia recording (download for 99¢ here Amazon.com) not that I’ve heard all the others of course.
Also used to similar effect for the opening of Aliens by James Horner who “interpreted” it virtually note-for-note without acknowledging the actual composer (Khachaturian).