There's a daily direct flight from Goa (GOI) to Mysuru (MYQ) by Alliance Air. This is the fastest (1Hr+) and most convenient option to travel from Goa to Mysuru. Flight prices hovers around ₹3500.

Otherwise fly to Bengaluru (BLR) and take the Bengaluru Airport to Mysuru Flybus.

There are no direct trains between Goa and Mysuru. Take a train to Mangalore from Goa (4 daily express trains) and take a Mangalore to Mysuru train (1 daily train) or bus. Other option is to take a direct train to Bengaluru from Goa and take a connection train or Bus to Mysuru (4hours)

Also watchout for special direct trains between Goa and Mysore. Indian railways operate special summer trains ( limited period)

KSRTC operate 2 direct buses to Mysuru from Goa's Panaji bus station. These are non-ac regular express bus (Sarige). Goa-Mysore bus fare is around ₹640. The journey is a bit hectic 16 hours. Probably this is the cheapest option to get to Mysore from Goa, if you are on a backpacker style trip & budget. Book ticket at ksrtc.in

If you are driving to Mysore, the most popular route is via Mangalore. Take the NH66 from Goa towards Mangalore. At Mangalore take the NH257 towards Mysore via Madikeri. Ther road distance is 600km and 12-14hrs drive.

For train you can reserve seats online at the www.irctc.co.in portal by the Indian railways; for KSRTC seat online reservation (KSRTC Booking) log on to the www.ksrtc.in
And of course please check with official portals for the accurate fares,schedules and other details.