Nice ! Looks like you're all set for your vacation ?
There are direct trains but no direct flights for this journey. kolkata (West Bengal) to Tiruchirapalli (Tamil Nadu) distance is 1663 km and you can get there by flight + flight in just over 6 hours and 45 minutes or by train fares starting from ₹ 616.
The heat there will make you sweat, but it's surely bearable.