There are no direct flights between Goa and Mysore.
From Goa (GOI) fly to Bangalore (BLR) there are a few daily direct flights from Goa to Bangalore. Goa-Bengaluru flight time is just over an hour.
The lone flight (SpiceJet) to Mysore (MYQ) from Bangalore is suspended. However check it out, there could be special flights scheduled during the festival seasons.
Also, if you are not able to get a direct flight to Bangalore from Goa, try the Goa-Mumbai-Bangalore sector. There there are quite a good number of options.