Though Mysore has an airport, there are no direct flights from Delhi. The best option is to fly to Bangalore from Delhi and continue onward to Mysore.
There are many dozen daily flights to Bangalore from Delhi.
From Bangalore to Mysore the flight operation has been suspended for some time.
Fly to Bangalore and travel to Mysore by Road or Rail. There are frequent train connections from Bangalore to Mysore.
Another alternative to look for one of those FlyBus services from Bangalore airport to Mysore City operated by KSRTC. See if the timings matches with your flight schedule. You can check the Flybus schedule and book ticket @ www.ksrtc.in. Opt for Source as Bangalore International Airport and Destination as Mysore.
Similarly Delhi to Mysore is connected by the train also.
Please check with the airline ticketing portals for connections and fairs. The ticket cost to Bangalore from Delhi vary anything from Rs5000 to say Rs7000 (approx $100-$180), or even goes up well beyond that during the peak seasons.
Location of Mysore on India map