I think this is one of the most underrated, underappreciated orchestral works of the Romantic age:
Josef Suk’s Scherzo Fantastique.
Seconded. Such a beautiful piece!
A lot of new things for me . . .
We’re definitely on the same wavelength - a number of you have recommended pieces that I’ve already ‘discovered’ and have been listening to (again and again) in the last months (e.g. the Franck Sonata and his Piano Trio, Mendelssohn Octet, Dvorak’s American, and Liszt, too).
Thank you all!
Ummmmm…concertos ARE symphonic. They are works for a lead instrument — most often piano or violin — with full orchestra. Okay, the Bee wrote a triple concerto (violin, cello, piano), and Brahms composed a superior double concerto (violin, cello).
EmilyG: Holy cats, another Suk buff! My favorite piece of his is the “Azrael” Symphony in C Minor for large orchestra. It’s some doom-haunted Scorpio shit, written shortly after the death of his wife.
Schubert’s eight piano impromptus are particular favourites of mine, but you may well already know them.
I’ll add the great Requiems by Brahms and Faure to your list.
Sticking with the singing, there has been no mention of opera yet - start with Puccini, then Verdi, and if you like those you might even get into Wagner.
Finally, don’t forget all the Lieder by Schubert, Schumann, Wolf, Strauss, et al.
For me, it would be because his 1993 recording of the late piano sonatas of Ludwig van Beethoven are a towering achievement in classical music. 
As to the OP, despite what you have written, I would still suggest giving Antonín Dvořák’s Symphony #9, From the New World a listen. Absolutely magnificent stuff. It’s extremely popular for a reason. 
Indeed, the New World Symphony is wonderful (but ‘familiarity breeds contempt’).
On the other hand, The Lark Ascending by Ralph Vaughan Williams was new to me. It is beyond words.
Funny thing, when I was a kid I would check out random classical albums from the library. I never checked out anything by him, because “Ralph” just seemed like a silly name for a serious composer. Then I found out it was pronounced “raif”, or “rafe”, so then I did. But I haven’t heard this, so I will indeed check it out now.
Dvorak’s Symphonies no. 7 and 8 are really good.
The Lark Ascending is nice, but Fantasia on a Theme of Thomas Tallis is transcendent, there is nothing else like it.
I know the OP was discounting symphonies, but Sibelius wrote several fantastic ones. Start with No. 2, and then No. 5.
And speaking of fantastic, there’s Berlioz Symphony Fantastique.
Also Tchaikovsky 5 and 6. And 4 is good too.
All these suggestions for symphonies are quite different than what you heard in the Classical Era or even among Romantic contemporaries.
As well as 1, 2 and 3. And his Piano Concerto #1 and Violin Concerto.
Seconded.
Tchaikovsky’s First Symphony is one of the most underrated works of the genre. It’s full of great brooding tunes and clever ideas from start to finish (ok, the finale may be a bit bombastic…). Anyway, I like a lot more than the highly-esteemed last three.
And Tchaikovsky’s Second Piano Concerto is quite underrated, obviously overshadowed by the First.
Here it is in Harpo’s words. Harpo Speaks! - Harpo Marx, Rowland Barber - Google Books
Harpo certainly was trying to annoy Rachmaninoff. But he wrote, “I didn’t really know until much later how sharp my intuition had been. I found out the great pianist and composer detested his Prelude in C-Sharp Minor.”
I’m assuming that if you’ve explored Schubert’s chamber music you already know his String Quintet in C major, which remains my all-time favorite of his chamber works.
At the other end of the time period I’d recommend Richard Strauss’s “Divertimento for Chamber Orchestra after Keyboard Pieces by Couperin”, which are delightful little pieces and not at all as heavy or harmonically dense as some of his better-known works. The third movement “Le Tic-Toc-Choc/La Lutine” is especially sweet.
A brief report:
A lot of folks mentioned various pieces by Mendelssohn. Thank you! They are wonderful.
In addition to the works recommended above, I discovered his Songs without Words (Lieder Ohne Worte). I am listening to it over and over again.
If you like Mendelssohn’s Songs Without Words, you might like Grieg’s Lyric Pieces.
Mendelssohn, you can’t go wrong with Mendelssohn.
Grieg, he wrote a lot of chestnuts. They sound good at first, but they weigh heavy on your ears after a dozen listens.
The classical flash mob in the Copenhagen subway eight years ago is a nice way to enjoy Grieg.
“Bach and Beethoven erected churches and temples on the heights. I only wanted… to build dwellings for men in which they might feel happy and at home.”
–Edvard Grieg.