Goa to Mysore distance is around 600km (372miles). That's roughly the road distance. The rail distances roughly around 684km between Mysore City and Vasco Da Gama in Goa.
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.
From Goa aim for one of these places : Londa Jn, Dhardwad or Karwar. There are quite a few direct trains from these stations to Mysore.
First let's look at the options from Karwar. There are a quite a few daily trains connects Karwar with Madgaon (MAO) station and Cancona (CNO) of Goa. It's about a hour's journey. You may not find a direct train to Karwar from Vasco Da Gama (VSO).
From Karwar board the Karwar Bangalore Express (Train No: 16524). This train leaves Karwar by 2.55PM and reaches Mysore Jn by 5.20 am next day.
The Route is : Karwar -> Mangalore Jn ->Sakleshpur-> Hassan -> Mysore Jn -> Bangalore City
This train continues from Mysore to Bangalore. The most scenic stretch in this route is the when the train crosses the Western Ghats (between Mangalore and Sakleshpur). But it would be in the mid night and you would not be able to see this. And there are not many trains passes through this stretch during the day time, except the Karwar to Yenvanthpur (Bangalore) Express.
See the Goa to Mysore train route map via Karwar.
You may also look for a suitable bus option to Karwar from Goa. Also see if you can prefer to do this till Mangalore and board a bus to Mysore. There are quite qood frequeny of buses from Mangalore to Mysore. Buses pass via Madikeri of Coorg, a very popular hill station close to Mysore.
Of course you can consider the reverse of this plan fro travel from Mysore to Goa. Karvar Express leaves Mysore Junction 23.50 pm and reaches Karwar in by 2.40 next day afternoon.
The second train route from Goa to Mysore is via Dharwad . 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.
From Dharwad board the Dharwad Mysore Express (Train No: 17302). This is a daily train between Mysore and Dharwad, leaving Dharwad by 8.30pm and reaches Mysore next day morning by 7.
There are also a few other Weekly / BiWeekly express trains between Dharwad and Mysore. The Sharavati Express (Train No:11035) is on Fridays . This is a day time travel to Mysore.
The Ajmir to Mysore Express (Train No: 16209) arrives on Mondays at Dharwad for Mysore. This is bit of circuits route, as the Ajmir Express heads to Bangalore before ending its journey at Mysore.
Another option is the Swarna Jayanthi Express (Train No:17302) that Connects Mysore with Delhi. This long distance weekly train arrives at Dharwad on Tuesdays 7.20PM and reaches Mysore next day early morning.
If you are doing the Goa-Hampi-Mysore circuit, consider the Hampi Express. This is a longish route for Goa from Mysore. If Hampi is in your itineary, this is an ideal train. Hospet (HPT) is the station for Hampi.
The Hampi Express connects Hubli (UBL) with Mysore via Bangalore. You can find a number of weekly trains from Vasco Da Gama to Hubli and also to Hospet.
Another very practical option is to book in the Vasco da Gama - Bangalore Link Express (Train No: 02779) up to Arsikere (ASK). This train leaves Vaso Da Gama by 3.10 PM and reaches Arsikere early at 3.45 am. From Arsikere travel to Mysore by the local Passenger train, leaves Arsikere at 5 AM and reaches Mysore by 9.25 AM. There are other options too (like the Gol Gumbaz Express Train no.16536 to Mysore) to travel from Arsikere to Mysore, the passenger train mentioned above is pretty convenient.
If the train routes looks a bit complex, look for one of those direct buses that operate between Goa and Mysore. KSRTC and Kadamba (both state owned) operate daily services. Also there are private operators.
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. For booking in private buses long on to the bus booking sites like www.redbus.in
And of course please check with official portals for the accurate fares,schedules and other details.