There are three options to travel from Mumbai to Mysore : by train, bus and flight.
Of course, people do drive all the way to Mysore by car, making it the third option.
The most convenient and economical means to reach Mysore from Mumbai is by train. There are a couple of direct trains (weekly/bi-weekly) that connect Mumbai with Mysore. If that is not a possibility, look for one of the many trains that connect Bangalore with Mumbai. From Bangalore to Mysore is well connected by road and rail.

There are two pairs of trains between Mumbai and Mysore.

Sharavathi Express (Train No:11035) operates on Tuesdays from Dadar (DR) station on Tuesdays to Mysore Junction (MYS).

Ajmer Mysore Express (Train No:16209) connects Ajmer in Rajasthan with Mysore. Catch this from Kalyan Jn (KYN) . This is a bi weekly train and you can catch it from Kalyan Jn for Mysore on every Saturdays and Mondays.

Another option is to split your train journey into  two. Here again there are two travel options. Option 1 is to travel to Bangalore from Mumbai and then get a connection (bus/train) to Mysore. There are multiple ways to reach Mysore from Bangalore. See also the connection train timings from Bangalore to Mysore.
The second option is to travel to Mangalore and get a connection train or bus to Mysore. These two are completely different train routes. See how to plan your  Mumbai to Mysore train journey.

Mumbai to Mysore by Bus : KSRTC operates a direct bus (A/c Volvo ) from Mumbai to Mysore. The bus leaves Mumbai ( Borivili , Opp. Hotel Gokul) by 3.0 pm and travels to Mysore via Shimoga. You can book KSRTC tickets online at their website (www.ksrtc.in).

And of course there are many luxury (Volvo A/C Multi Axle Semisleeper type ) private  bus services too between Mysore and Mumbai (like VRL Travels, SRS Travels , Kallada Travels etc).

Log on to one of those bus ticket booking websites like redbus.in, makemytrip.com etc for fare anquiry and booking. Bus fare hovers around Rs1500 to Rs 2000 for volvo type bus. Can shoot up to even Rs4000+ during the peak demand time.
Mumbai to Mysore Flights: 

About the air connectivity, Mysore is connected to the rest of major airports in the country essentially via Bangalore.

Though Mysore has an airport, regular flights are not operated. SpiceJet, which used to fly from Chennai to Mysore via Bangalore as stopped their operations to Mysore.

Fly to Bangalore from Mumbai and then travel to Mysore from Bangalore or take that Bangalore airport to Mysore Fly Bus.

A word of caution. Use the details provided here for your basic reference. Before taking travel decisions please consult more authentic and up to date official sources (irctc.co.in etc). Do this especially for the schedules, as there could be changes, which might not be updated here.