Whoa - that looks like an exciting journey to take !
There are direct trains but no direct flights for this journey. Mumbai (Maharashtra) to Tiruchirapalli (Tamil Nadu) distance is 1344 km and you can get there by flight + flight in just over 6 hours and 5 minutes or by train fares starting from ₹ 515.
The heat there will make you sweat, but it's surely bearable.