Hmm .. that journey is surely more than just a weekend trip ?
There are direct trains but no direct flights for this journey. kolkata (West Bengal) to Ajmer (Rajasthan) distance is 1642 km and you can get there by flight + cab in just over 7 hours and 42 minutes or by train fares starting from ₹ 536.
The heat there will make you sweat, but it's surely bearable.