My tastes also tend towards Romantic composers and later; earlier music tends to be a lot drier and less sanguine. But I still think it's worth making a case for more specific examples from before Beethoven. 3 off the top of my head...
Allegri - Miserere mei, Deus
When performed by a good choir this is sublime.
Bach - The Brandenburg Concertos
Bach is the hands-down greatest composer of all time, and omitting him from a musical diet is like omitting chick peas from a vegan diet. The 6 of these concertos together are really worth seeing. The insane harpsichord cadenza/solo in the 5th concerto is awesome, especially if you take a look at the score.
Mozart - Requiem
I tend to dislike Mozart more than other Classical composers, and I think it's mainly because it largely sounds immature and frivolous to me (he did die young after all). However, the requiem was written in the year of his death, and it is as dark and intense as he gets.
The most accessible 19th and 20th century pieces I'd recommend for absolute beginners:
Stravinsky - Rite of Spring
Orff - Carmina Burana
Rachmaninov - Second piano concerto
Gershwin - Rhapsody in Blue
Beethoven - 9th symphony
Gorecki - 3rd symphony
Elgar - Cello concerto
I love the Rachmaninov, Sibelius, Mahler, Bruckner and Shostakovich mentioned already, and in addition I think it's worth checking out:
Bruch's violin concerto
Albeniz Chants d'Espagne and Suite española (either of which is great on piano or guitar)
Strauss' four last songs
If you like Chopin, I recommend the Etudes which are immensely virtuosic and richly varied, and not in the least esoteric, though they are not performed so often because they're so difficult.
Opera-wise, I think the easiest to get into are
Bizet - Carmen
Mozart - Magic Flute (more fun and interesting than his others, which are all a little long and repetitive for my taste)
Verdi - Aida
I also massively recommend Wagner, but spending 6 hours in the same seat is in itself an acquired taste
Nothing specific planned at the moment, but I tend to go less to big orchestral events and more to chamber performances or recitals, where you only know the performer beforehand (usually students or youngsters with potential), and rarely the repertoire. However, I did recently see the Sondheim night at the Proms which I loved. I was going to get to a couple of other proms this season, but seem to have run out of time