Pack some nice homemade food, cause this isn't a short trip !
There are direct trains and buses but no direct flights for this journey. Mumbai (Maharashtra) to Mysore (Karnataka) distance is 1045 km and you can get there by flight + flight in just over 5 hours and 35 minutes or by train fares starting from ₹ 445.
The weather is amazing right now. You'll have a great time.