There are three popular options to travel from Bangalore airport to Mysore. Mysore City is about 190km southwest of Bangalore airport. You'll need to cross Bangalore city, or the outskirts of it, before heading towards Mysore city.
Travel Option 1. By Train
Travel Option 2. By Bus
Travel Option 3. By Cab/Taxi
Travel Option 3. By Car (self-driven)
Bangalore Airport to Mysore by Train: First you've to get to Bangalore City railway station (SBC) either by bus or by the airport taxi. Then get into one of those Mysore bound trains (Bangalore to Mysore by train).
There are quite frequent bus connections from Bangalore airport to Bangalore city center (Majestic) where the Bangalore city railway station is located. The bus stops at the city bus station across the road from the railway station, it's a 10 minutes walk to the railway station.
Bangalore airport to Bangalore city travel time is about 1+ hour by taxi, close to 2 hours by the airport bus. The train takes another 3 hours to Mysore.
Also keep in mind there is a nearby railway (20km) station Yelahanka to the Bangalore airport. Some of the long distance Mysore bound trains stops at Yelahanka station (like Hampi Express, Basava Express). See if you can get a connection. You'll save good deal of time traveling to the city center.
Bangalore Airport to Mysore by bus: There are 4 direct buses everyday operating from Bangalore airport to Mysore, called Flybus. By far this direct Bangalore airport to Mysore bus service, is the best option to travel directly from Bangalore airport to Mysore.
Another alternative is to get to the Mysore Road Satellite Bus Station in Bangalore city from airport. Further you can get into one of those Mysore bound buses (Bangalore to Mysore by bus). Again you've to depend on the airport bus or taxi services to get to the Mysore Road Satellite bus station. It's about 2 hours travel, towards the direction of Mysore.
Bangalore to Mysore bus (KSRTC) travel takes about 3 hours further.
Bangalore Airport to Mysore by Cab/Taxi: This is a convenient, but expensive option. Book in advance with a reputed cab operators who offers a meet and greet airport pickup, rather than booking one at the airport.
The road distance from Bangalore Airport to Mysore city is about 190km. Taxi fare varies depending on the package (drop vs return vs sightseeing etc ) .
There are those online services (OLA, Meru, TaxiForSure etc) as well as the regular tourist cab operators (Bharath International Travels, KSTDC etc ). Check for the rates and pick the package suits your situation. Avoid booking through touts.
Bangalore Airport to Mysore by Car: If you prefer, Bangalore also has self-driven car rental service (zoomcar.com) . You can pick it up from airport and drop back on return from Mysore.
To drive to Mysore, you need to go around (or through) Bangalore city to get to the Mysore highway.
Take the NH7 from airport towards the city. Exit NH7 at Hebbal Flyover (26 km from airport) and take the right diversion into the Outer Ring Road (ORR). Continue on the ORR till it joins NH4 (aka, Tumkur Road). Drive towards Tumkur till you see the entrance to the NICE Expressway (toll road).
Get into the expressway and take the Mysore Road exit. You are now practically out of Bangalore and on the way to Mysore. See the Bangalore to Mysore by road for further driving directions.