Classical music newbie needs your suggestions

There are lots of good suggestions above which I will try not to repeat. Here are some I highly recommend:

• Bach - Brandenburg Concertos
• Bach - organ music, especially Toccata in F (BWV 540), Prelude & Fugue in A minor (BWV 543), and Prelude & Fugue in D major (BWV 532). Since this is an audio forum, you have to test your woofers with some really low bass, and only the pipe organ can really do that (16 Hz). 😁
• A Bach Festival, Empire Brass and Douglas Major, organ
• Music for Organ, Brass, and Percussion, Empire Brass and Michael Murray, organ (Telarc).
• Handel's Messiah
• Saint-Saens - Symphony No. 3
• Charles-Marie Widor - Symphony 5 for organ
• Rachmaninoff - Rhapsody on a theme of Paganini, all four piano concertos, 24 preludes for piano.
• Holst - the Planets (especially the Charles Dutoit/Montreal Symphony performance).
• Basically anything by Tchaikovsky.
• Sibelius' violin concerto.
• Smetana - Ma Vlast, especially the Moldau.

I'm sure I can think of more if I look at my CD collection, but that's a good start. Have fun!  Some of these will take multiple listens to really understand, but they're all great music.

You might enjoy owning Mahler’s 2nd symphony "Resurrection" conducted by Gilbert Kaplan. The story behind the man’s determination to conduct a symphony orchestra is rather interesting.

See

I've been into classical music for decades and love it. Some personal favorites include pianist Sviatoslav Richter, and the Guarneri Quartet, especially their late Beethoven set that came out on Philips. A whole level beyond anyone in insight for me. Sandor Vegh and the Camerata Salzburg have a 10-CD set of Mozart serenades and divertimentos that's great. Bernstein's Schubert 9th and Unfinished (#8) symphonies with the Concertgebouw are special. Haydn quartets with the Auryn Quartet are wonderful.

Yes I enjoy the Guarneri Quartet also. Especially Mendelssohn string quartet #3. 

Don't forget Vladimir  Horowitz, the greatest classical pianist of all time and the best natural pianist of all time Glen Gould who said he only practiced once per week and when he was asked what he thought of Jazz, which I enjoy, he said it was a waist of some very good notes.