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