Both destinations are served by Dhaka (DAC). Since there's no flight route between them, the best way to travel is by road.
Barrackpore
Fly to Dhaka (DAC), then travel by road to Barrackpore. Taxis, buses, and private cabs are available from the airport.
Dhaka
Fly to Zia Int'l (DAC), then travel by road to Dhaka. Local transport options include taxis, auto-rickshaws, and state buses.