Both Cynthia and Bob repeatedly refer to Bob as a piccolo player, but in orchestra scenes he is mostly shown playing the flute. However, it is not unusual for flutists to switch to the piccolo when a piece requires it.
In Hailey's oboe lesson, Betty says [about the music], "Vaughan Williams called it a rondo pastorale. But you're, like, clear cutting an entire forest." This is a joke on the musical genre: a pastorale is a light, gentle mood, inspired by peaceful nature/woodland scenes, while a rondo is a lively, often folk-inspired, dance form with repeating themes. Betty's insult means that Hailey plays with no delicacy or nuance.
When Rodrigo and Ana Maria discuss the Sibelius violin concerto, he says "the Adagio, followed by the Allegro ma non troppo." He is describing the tempo markings for the second and third movements. But this is also a sex joke, not only because the Sibelius is the piece they "first made love to," but because the tempo markings mean "slow" and "fast but not too fast," respectively, in the original Italian.
In the opening scene, Ana Maria plays the violin left-handed - the opposite of standard playing. Most violinists place the instrument on their left shoulder, use their left hand on the fingerboard, and bow with their (dominant) right hand. Ana Maria's reverse playing means that she is left-handed, and that her violin's strings are strung in the opposite order than normal.