Nakorn Si Thammarat is a relatively easy place to get to, although it is a bit off the main north-south travel routes.
Nakorn Si Thammarat's small airport is served by at least two flights a day from Bangkok. One flight is operated by the national carrier Thai Airways, although they are now letting their budget carrier, Nok Air, handle the route.
Nakorn Si Thammarat is on a spur off the main north-south line. This means that all trains heading south from Bangkok are not necessarily headed for Nakorn Si Thammarat. There are, at last check, only two trains a day that terminate in the city.
Like the rest of Thailand, inter-city buses are available between Nakorn Si Thammarat and most anywhere else. Your choices are both air conditioned and non-air conditioned buses.