Generally, Tamil Nadu is one among the safest places in India. But like any other tourist destination, there are a few problems. Beware of pickpockets and thieves in Buses and Trains. If you travel via Train, think twice before eating anything offered by co-passengers.
Also, when traveling with someone from the opposite gender, it would be advisable to avoid too much of physical intimacy, as it may invite unwanted attention and comments.It is also best if you do not talk about the Sri Lankan Civil War between the mainly Sinhalese Government and Tamil fighters.
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Tamils are fiercely proud of the Tamil language, one of the few still living classical languages, with two millennia of written tradition. Unlike many other Indian languages, it belongs to the Dravidian language family that includes the other major South Indian languages - Telugu, Malayalam and Kannada. Tamil Nadu was the site of violent anti-Hindi riots in 1965 when the union government tried to impose Hindi as the sole official language of India, and there is still some residual hostility towards Hindi, so try to pick up some instant Tamil here. Even if you are in Chennai, do not try to speak to locals in Hindi, as many will find it offensive. English works in almost all the cities.
However, if you don't speak Tamil, most educated Tamils speak English fluently, while many others know some basic English, so you shouldn't have a problem getting by with English in the major cities.