Driving to the chateau is pretty easy from Paris! You can put Le Chateau de Thauvenay into your GPS, and you’ll arrive in about 2 hours. We do not recommend renting a car in the city center (as driving there can be quite hectic), but instead renting from the airport or just outside the city proper. 
Taking a train is also pretty easy! There are a few trains everyday departing from Paris Gare de Bercy (12th Arrondissement) and arriving at Tracy - Sancerre, which is just 5 minutes from the chateau. The fastest trains should get you there in less than 2 hours, and tickets are usually around 30-40 Euro. There are also additional trains that go to Cosne-sur-Loire, which is about 20 minutes from the chateau. We can see if anyone is available to pick you up, or cabs / Ubers /FreeNow should be available! You can purchase train tickets here: https://www.sncf-connect.com/en-en/train-ticket. 
BlaBlaCar is also an option if you don’t want to rent a car or take the train. A few days before your departure you can search to see if other French drivers are making a trip nearby and you can arrange to ride with them.