Mysore to Goa distance is about 600km. The shortest route is via Shimoga, while the route via Hubli & Karwar should quite fast.
Route 1(Mysore-Goa(Panjim) distance:630km ): Mysore --> Madikeri --> Mangalore --> Honnavar--> Panjim
Route 2(Mysore-Goa(Panjim) distance:600km): Mysore --> Channarayapatna --> Arsikere--> Shimoga --> Sagar--> Honnavar --> Panjim
Route 3(Mysore-Goa(Panjim) distance:650km): Mysore --> Nagamangala --> Turuvekere --> Hiriyur --> Chitradurga --> Davangere --> Haveri--> Hubli-->Dharwad -- > Ponda -->Panjim
Route 4(Mysore-Goa(Panjim) distance:730km): Mysore --> Nagamangala --> Turuvekere --> Hiriyur --> Chitradurga --> Davangere --> Haveri--> Hubli -- > Karwar -->Panjim
Mysore-Goa Route 1: Take the Hunsur road (SH88) out of Mysore City towards Madikeri. At Madikeri town, take the diversion to Mangalore. From Mangalore take diversion to join the NH17, that pass through Goa. After Karwar you'll ener south of Goa. Select this route, if your destination in in the southern parts of Goa. Distance from Mysore to Panjim via this route is about 630km.
Mysore to Bilikere is 4 lane, after till about Mangalore it's 2 lane highway. At Mani the SH88 joins the NH48 connecting Bangalore with Mangalore. The road around the Madikeri stretch is a bit curvy, thanks the hilly terrain.
Mysore - Goa : Route 2: Take the Mysore to Bangalore road (SH17) out of Mysore City. After you cross Srirangapatna, you'll have to take a left diversion to Pandavapura where you'll cross a railway track. After crossing the rail track, teke the left diversion towards KRPet, Channarayapatna and Arsikere. At Arsikere town , take the left diversion on to the NH206. The highway further cross Shimoga, Sagar and crosses the Ghat section before joining NH17 at Honnavar. From here onward, the route is same as that of Route 1.
Mysore-Goa Route 3: Take the same route till Pandavapura, as in Route 2. After the railway cross at Pandavapura, take the right diversion to Nagamangala. At Nelligere you'll cross the NH48 and continue towards Turuvekere.
Further at Hiriyur you'll hit the NH4. Continue on NH4 till Dharwad. At Kelageri take the diversion to Londa. The route is through forest before you enter Goa. Take this route only if you are very keen to drive through the forest.
Mysore-Goa Route 4: This route to Goa is essentially the same as Route 3 till Hubli. At Hubli bypass take the left diversion (NH63) towards Akola where NH63 joins the Goa bound NH17. Though distance wise this may appear a lot longer, say compared to Route 2 to Goa, A great distance of this route is through the 4 lane NH4. Also on this route there is a shorter diversion to hit (Bankapur->Mudgod->Yellapura) .
Keep in mind, while this diversion appears shorter,but you will be drive slow thanks to the many villages en route and also this shortcut route is not as great as the highways.
Another option is to continue till Belgaum and take diversion towards Goa.