The city’s two major opera companies occupy adjoining buildings in Lincoln Center. The Met may have the glamour and the Italian tenors but City Center can lay claim to a more modern and adventurous repertory.
Both are superb. As for concerts of classical music, you’re likely to find 30 or so scheduled for a single evening with the New York Philharmonic (also at Lincoln Center) heading the bill.
Carnegie Hall, known for its excellent acoustics, plays host to outstanding visiting artists. Free concerts are held frequently at Lincoln Center’s Library of the Performing Arts and at the Juilliard School of Music. Summers, the Met and the Philharmonic give free performances in the city’s parks.