First of all,its safe travelling in train or buses in TN except for late hours,
You can reach Chidambaram from Chennai by Bus or Train or Taxis
1.Bus : Go to Chennai Mofussil Bu stand in Koyembedu (45 mins from Chennai airport),plenty of buses are available Chidambaram ,it takes 4 -5 hrs in bus,since Pondicherry and cuddalore falls in way to chidambaram , you can all take buses to these places and make transit journey.
2.Night trains are available from Chennai Egmore station to Chidambaram
1, you can board from Egmore station or even Tambaram station in Chennai
3. Taxi from airport will take you 4 hrs to chidambaram