The city's name was Dayong until 1994, when it was renamed to link it to the Zhangjiajie National Forest Park (itself named after a tiny village now known as Zhangjiajie village or Zhangjiajie Cun).
Zhangjiajie is about 250 km (155 mi) northwest of Changsha.
The local airport is Zhangjiajie Hehua Airport (DYG IATA; formerly named Dayong Airport). It has direct flights to many domestic locations, as well as Taiwan and South Korea. Flights are relatively scarce - about 25 per day total.
Be careful with the taxi drivers in the airport parking lot. They are likely unwilling to use the meter. They may claim that your destination is very far and attempt to charge you a "parking lot fee". Call your hotel or hostel to find out how much the trip should cost and ask them to speak with a taxi driver for you. It should cost only ¥20-40 total for a trip from the airport to Zhangjiajie city.
You can take a train from Changsha, about 4 hours, or Huaihua, about 4 hours. Train from Shenzhen West (K0964) to Zhangjiajie, 17 hours, overnight, about ¥480.
Intercity buses from Changsha take 5 hours, and are boarded at the Changsha West bus terminal (a fancy new building which resembles a shopping mall, next to the Wangchengpo subway stop). These buses run approximately hourly with the last bus departing around 1900. Fare is about ¥120.
Taxis are cheap so they are probably your best bet within the city. A bus can take you out to Wulingyuan.
Zhangjiajie Central Bus Station operates mini busses that will take you next to Zhangjiajie National Forest Park's entrance. You do not need to buy a ticket at the Central Bus Station ticket window, simply go thru security, keep to the left and make your way outside thru Exit 1. The bus fare to the park is ¥12, and you pay the driver directly. Keep in mind that these are older mini busses which are used primarily by local commuters to their jobsites. Many stops will be made picking up and dropping off people along the route. The terminus appears to be a partially constructed hotel or apartment complex about five minutes on foot the park entrance. The trip is about 1hr 20min, and the roads are substandard as of Jan 2018. Once you are done in the park, return to the location you were dropped off at. Busses will be waiting to return you to the Central Bus Station. The mini busses depart in both directions about every 20-30 minutes. The first bus leaves the Central Bus Station at 06:30 the last one returns from the park around 18:00.
There are limited food options in the park, and what is available is offered at highly inflated prices. It is advised that you bring food and water with you to the park.
Accommodation in Zhangjiajie is typically around ¥170. There are many options within walking distance surrounding the train/bus station.
Though Tianmen Mountain and Wulingyuan are major tourist attractions, perhaps 99% of their visitors are native Chinese. So you will find it surprisingly difficult to change foreign currency. There appear to be no currency exchange stores. Perhaps the only place to change money is a bank, such as the Bank of China (follow the main road north of the bus/train station area, it is at the first major intersection north of the river, about a 15-minute walk from the bus/train stations).
Wulingyuan, however, is rumored to have currency exchange stores.