Strictly speaking, the Classical period in Western music, corresponding to the neoclassical in the other arts, was quite brief – most precisely, 1770–1820. But in common usage, "classical" refers to music that has stood the test of time, music that belongs to a continuing repertory.
At the most basic level, you can listen for pure sonic pleasure. For this I would suggest Stravinsky's Firebird, Debussy's Prelude to the Afternoon of a Faun, or Rimsky-Korsakov's The Golden Cockerel, music in which orchestral color, or timbre, bears as much importance as the notes themselves.
You can listen for melody. For some composers, the tune is most important. Listen to Grieg's Last Spring or a Puccini aria.
You can listen for structure. The four most basic structures in "classical" music are ABA, or song-form; theme and variations; sonata form; and rondo. ABA features long lines and a return to the original material; listen for the subtle ways in which the composer alters the return. In theme and variations, marvel at the composer's ingenuity. How many different ways can the theme be presented; how much can it be changed while still retaining its essence? Sonata form presents musical ideas, creating in the listener a sense of home, then chops up the ideas, subjecting the fragments to different keys, disorienting the listener. When the ideas are reassembled in the home key, great is the listener's relief! Rondo form features many returns of a cheerful melody, each return greeted with delight. Humans are "meaning makers" and listening for structure helps us bring order to the chaos.
It helps to know something about the composers as it enhances the listener's experience. For example, Beethoven was the first autobiographical composer. The struggle to "overcome the darker powers" that characterizes the music of his middle period reflects his personal struggle, a human struggle. Different composers must be approached differently. Consider Tchaikovsky and Brahms, two contemporaries from the Romantic period: Tchaikovsky's music is immediately appealing, emotional and visceral, while the music of Brahms takes multiple hearings in order to reveal its wondrous craft.
Finally, you can listen for story. Music tells a story, not in words but in sound. The narrative takes place over time; it is a journey from the first note to the last. Listen to the story; take the journey – it is a journey of self-discovery.