Whoa - that looks like an exciting journey to take !
There are direct trains but no direct flights for this journey. New Delhi (Delhi) to Mysore (Karnataka) distance is 2288 km and you can get there by flight + flight in just over 6 hours and 45 minutes or by train fares starting from ₹ 702.
The weather at Mysore is varying from "nice" to "hotter than nice".