There are no direct trains between Mysore and Goa. However if you can split the journey into two laps, you can manage to get some decent connections.
Change at Londa Jn (LD): Take any train/bus to Londa Jn located east of Goa (3½ hours). The daily Vishwa Manav Express (Train No: 17325) can take you to Londa Jn to Mysore Jn via Bengaluru. Other weekly trains and the Ajmer-Mysore Express (Train No: 16209) and Sharavathi Express (Train No: 11035)
Change at Dharwad (DWR): Take the daily Mysore Express (Train No: 17302) . This is an overnight train to Mysore.
Dharwad is about 180 km east of Vasco Da Gama. There are a couple of daily trains and a few more weekly/biweekly/triweekly trains connect Vasco with Londa Junction & Dharwad.
The Sharavati Express (Train No:11035) is on Fridays . This is a day time travel to Mysore.
The Ajmer to Mysore Express (Train No: 16209) arrives on Mondays at Dharwad for Mysore. This is bit of circuits route, as the Ajmer Express heads to Bangalore before ending its journey at Mysore.
Another option is the Swarna Jayanthi Express (Train No:12782) that Connects Mysore with Delhi. This long distance weekly train arrives at Dharwad on Tuesdays.
If you are doing the Goa-Hampi-Mysore circuit, consider Hampi Express. Hampi Express connects Hubli (UBL) with Mysore via Bangalore. This is a longish route for Goa from Mysore. If Hampi is in your itinerary, this is an ideal train. Hospet (HPT) is the station for Hampi.
You can find a number of weekly trains from Vasco Da Gama to Hubli and also to Hospet.
For trains, 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.