There are auto rickshaws, but i found it difficult to bargain with them. Hence, i preferred taking the bus around. The easiest way would be to head to Vadachery or Nagercoil bus stands. There are a number of buses going from there to different places. You could ask any local which bus to get into.
There are a number of trains running to Nagercoil Junction from various parts of the country. Most trains coming here would come through Madurai Junction. This would be the easiest way to get there.
Alternately, one can take trains to Trivandrum. Trivandrum is a couple of hours from Nagercoil.
Kanyakumari is another train station close by.
During our visit to Kanyakumari, we hired a car for the day and while paying a visit to Padmanabhapuram, we crossed the town of Nagercoil which lies at a distance of only 19 kilometers from Kanyakumari. It will hardly be an hours journey from Kanyakumari.