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