There are two direct trains between Mumbai and Mysore, the Sharavati Express and the Ajmer Mysore Express. These are weekly /biweekly express trains.

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

It takes about 24 hours by Sharavathi Express to reach Mysore. This train leaves Dadar by 9.30 in the night and reaches Mysore railway station around the same time next day.

The second direct option is the long distance Ajmer Mysore Express (Train No:16209) that connects Ajmer in Rajasthan with Mysore. You need to get to Kalyan Jn (KYN) to catch this train.   The train leaves Kalyan Jn (Saturdays and Mondays) by 1.10 past midnight and reaches Mysore after 7 in the morning, making it a 30 hour journey between Kalyan and Mysore.
 About the return routes, Sharavati Express ( Train No:11036) leaves Mysore Junction every Thursdays by 6.15 in the morning. On the other hand the Rajasthan bound Ajmer Express (Train No:16210) leaves Mysore on every Tuesdays and Thursdays 6.15 in the evening.

Both  Ajmer Express and Sharavati Express follow more or less the same route between Mumbai and Mysore up to Arsikere Junction (ASK).

From there Sharavati Express takes a southward direct route to Mysore via Hassan. On the other hand Ajmer Express travel via Bangalore which is a bit longer route. Otherwise the route between Mumbai and Mysore is via Pune, Belgaum and Hubli.

If the direct trains are not an option for you try to split the journey into two. In this case you can chose between two major train routes between Mumbai and Mysore. The first route is Mumbai-Bangalore- Mysore and the second route is Mumbai - Mangalore- Mysore.

Mumbai to Mysore via Bangalore: 
There are at least three daily express trains and many weekly/bi-weekly trains connecting Bangalore with Mumbai.

For example Udyan Express ( Train No:11301), Coimbatore Express (Train No:11013), Chalukya Express/Puducherry Express ( Train No:11005) and so on.

Find out the railway stations for Mumbai and Bangalore for each trains. On the Bangalore side if possible, prefer the trains that take you to Bangalore City (SBC). It is much easy to get a connection train for Mysore from Bangalore City railway station, or if you have to travel to Mysore by bus, the KSRTC bus stand is just across the railway station. Bangalore to Mysore by bus or train is about 3 hours journey.

Mumbai to Mysore via Mangalore: There are quite a few express trains in the Mumbai to Mangalore route (Konkan rail route) . Matsyaganda and Netravathi Express are daily trains. There is also a Shatabdi Express towards Mangalore from Mumbai.

From Mangalore look for a connection train to Mysore. The Mangalore-Mangalore Express (Train No 16518 or 16525) is a night train to Bangalore from Mangalore via Mysore.

There are also buses between Mangalore and Mysore.

You may also want to see the flight and bus options from Mumbai to Mysore.

The above train details could have changed and not updated here. For the updated and current train schedules, check with one of the official sources (like irctc.co.in).