New to classical music

For eclectic I would also suggest Arvo Part.

@elizabeth , two really great recommendations; Beethoven
and Rachmaninoff piano concerti.
 But I have to strongly disagree with your opinion of Mahler. I would say that it's not for beginners.

Here are some of my favorites that a listener who appreciates the eclectic might appreciate:

Alan Hovhaness - Symphony #50 "Mount St. Helens"
"And God Created Great Whales" Opus 229

Olivier Messiaen - "Des Canyons aux Étoile"
"Turangilila Symphony"

Bejamin Britten - "Prince of the Pagoda"

Colin McPhee - "O Bali"

Igor Stravinsky - "The Rite of Spring"
"Petrushka"
"The Firebird"

Bela Bartok - "Concerto for Orchestra"
"Music for Strings, Percussion, and Celesta"

Hector Berlioz - "Symphony Fantastique"

Einhojuhani Rautavaara -"Cantus Articus "Concerto for Birds and Orchestra."

All of these works showcase the power of the orchestra in different ways.

for piano - any well recorded, well performed Chopin.  

Baroque - Vivaldi and Bach

Classical - Beethoven (yes, he really was a genuine genius). Any Mozart (very tuneful and pleasant to hear)

Romantic - any of the Russian composers, specifically Tchaikovsky, Rimsky-Korsakov, Prokofiev, Stravinsky, Glazunov

Impressionism - Ravel and Debussy

Once your tastes mature, Shostakovich.  His symphonies #1, 4, 5, 6, 7, 8. 9, 10 and 11 especially.

If you want you can try Mahler and Bruckner but I just find it cold and sterile.  Just not what reaches my soul.  Go for Shostakovich if you want to experience a panoply of human emotions.

Good luck on your journey!