Handel : Great suggestions so far, so I’ll only add the logical follow-up i.e. …
Bach. Try the sacred vocal works, the dramatic and gorgeous St Matthew Passion first and foremost. I’d advise against listening to the whole 3-hour piece initailly. Find some exerpts which must however include the opening and closing choruses (Kommt, ihr Töchter, helft mir klagen + Wir setzen uns mit Tränen nieder, respectively) as well as the aria Erbarme dich. Want more ? Go for the extatic Magnificat or some of the cantatas (Ich glaube, lieber Herr, hilf meinem Unglauben, BWV 109 + Herz und Mund und Tat und Leben, BWV 147 are personal favourites but there are almost 200 to choose from…).
If you prefer instrumental works, you could check out some of the solo harpsichord/piano pieces in the Well-Tempered Clavier (Book 1, preludes and fugues 1, 2, 4, 16 for a start), the flute sonatas (BWV 1030 first), the cello suites (1 is the most famous one), the orchestral suites (2 and 3) or the violin concertos.
Beethoven : his piano sonatas are essential, all 32 of them. Ok, more realistically, start with the famous ones (8 “Pathetic”, 14 “Moonlight”, 15 “Pastoral”) and then move on to the introspective, metaphysical last ones (30, 31 and especially 32). Once you’re hooked, you’ll want to know them all. You should also give his piano concertos (4 and 5) and symphonies a try (at least the fifth !). And, as jayjay mentioned, his Great Fugue, too.
Logical follow-up : Brahms ! First and foremost his late piano pieces op. 116-119 as well as his clarinet quintet, all of these are achingly beautiful, the works of a composer in the autumn of his life who masters his art totally. Wistful but never maudlin. Then, his chamber works (the first cello sonata, the third violin sonata), symphonies 4 (!) and 1 as well as the first piano concerto
Ravel. The piano concerto in G, with one of the most beautiful slow movement in the whole repertoire. His solo piano cycle “Miroirs” is also wonderful as well as his vocal works : “Shéhérazade”, “Chansons Madécasses” and “Trois Poèmes de Stéphane Mallarmé” (the latter is perhaps Ravel’s most experimental piece, so leave it for later).
Logical follow-up : Debussy (actually the oldest of the two composers). The piano works (Préludes, Images and Estampes + Six Epigraphes Antiques for two pianos) and the orchestral pieces (La Mer, Prélude à l’Après-Midi d’un Faune, Trois Nocturnes).
If you still have some energy left (I hope my suggestions were not overwhelming), try Chopin’s heartbeaking Nocturnes and Tchaikovsky’s violin concerto.
I excluded almost all 20th-century composers as many are an acquired taste but some are well worth discovering, too.