Chennai Airport is one of the busiest airports in the country connecting Tamil Nadu to rest of India. You can connect to south-east part of Tamil Nadu by taking a flight till Chennai and then taking a train or bus to these places. To connect to the western areas you can take a flight till coimbatore and then take a bus or train to the destination. To the south most tip, like Kanyakumari, Tirunelveli, Nagarcoil etc..it is better to take a flight connection till Trivandrum (Kerala) and from there take a bus or train.
The railway hub of Tamil Nadu is Chennai (Madras) and Coimbatore. These stations connect the state to all the other states of India. These are also major hubs connecting Kerala (Via Coimbatore) and the south-east of Tamil Nadu (Via - Chennai). There are trains running on a daily basis connecting Chennai and Coimbatore to various states across the country.