There are many buses connecting Mysore to cities such as Mumbai (1013 km), Pune (863 km), Nashik (1076 km), Chennai (478 km). The best option is to travel from Mumbai via Expressway and Tumkur Rd. It is approximately 15-16 hours to Mysore Palace. The most economical option to travel to Mysore from Bangalore is to board an air-conditioned Volvo Bus that departs from the Mumbai City Bus Stand.
You can take a train from any of the metro cities – New Delhi, Chennai and Bengaluru (Bangalore), and Mumbai. There are two pairs of trains between Mumbai and Mysore one is Sharavathi Express and Ajmer Mysore Express. The Mysore Palace is about 3kms from the Mysore City Station and just 1km away from United-21 Hotel Mysore.
Mysore at present does not have an Airport, although work was under way to upgrade the Mandakalli Airport near the city. You need to fly in to the Bengaluru International Airport, which is the closest airport to Mysore. From here, you can either take a train or travel by road.