Chidambaram to Tiruvannamalai direct bus frequency is less. Few long destination buses Tirupattur, Tirupathi & Vellore will go through Tiruvannamalai.

If you could not find one, change over buses at Cuddalore / Villupuram. There is a daily passenger train runs from Chidambaram to Tiruvannamalai,.(Original start and end points are Mayiladuthurai and Tirupathi) You cannot find this info in any website. It is 9 coach unreserved train.

Dep time from Chidambaram : 16:30. Arrival time Tiruvannamalai - 20:30.

Check the time table of the two different train numbers below. But it is the same train. You dont need to change over at Villupuram.

https://indiarailinfo.com/train/timetable/mayiladuthurai-villupuram-passenger-56876/21805/2755/785#st

https://indiarailinfo.com/train/timetable/villupuram-katpadi-passenger-56886/21813/785/6917#st

in chidambaram there is thillai nadarajar temple as you already known. for visiting any ashram you can prefer tiruvannamalai like ramana maharishi ashram and lot others. there are lot of temples near to chidambaram too like sirkazhi temple, gangai konda cholapuram and kumbakonam. if you dont have time and doesnt like to add you can take a bus directly from chidambaram to tiruvanamalai. there is no timing. but buses are there without timing. even any buses to vellore and tirupathi will go via thiruvanamalai. if doesnt want to wait then take a bus to panruti or villupuram and change over from there. by this way you will lot more of buses. there is no train link between the two towns. for train you have to go to villupuram and change over to bus to tiruvanamalai