a) To soundproof the ceiling, add fiberglass batts without a vapor barrier between the joists. Then fasten one or two layers of drywall to them, using resilient clips and metal furring channels (called hat channels, for their shape). This isolates the drywall from the joists, eliminating vibration and thus minimizing sound travel.
b) Place mechanical equipment on top of sound-dampening antivibration mats or pads.
c) Minimize connections between ducts in the basement and upper floors to reduce noise transmission.