There's no train connection with Chengdu Airport.
Car and bus
Chengdu Shuangliu Airport is situated 16 kilometers southwest of Chengdu's city center. The airport is easily accessible by rental car via the ring road G4201. From the south of the city the Airport Expressway (a toll road) leads to the airport.
City bus 303 runs from the airport to the northern train station in Chengdu. Shuttle busses also run every 20 minutes between the airport and Chengdu with stops at the Minhan Hotel and the Lantian Hotel in the city center. A bus tickets costs 10 yuan. Traveling time is about half an hour.
Car rental Chengdu Shuangliu Airport
There are no international car rental companies present at Chengdu Airport but there are counters of local providers. International rental companies, such as Avis, do have an office in the city.
There are taxis outside the Arrivals Hall. They don't have a meter and if they do it's broken or the driver refuses to use it. You'll therefore have to negotiate. A reasonable price for a ride into the city is 50 yuan, but the asking price by the driver will initially be significantly higher. A ride to Chengdu's center takes 20-30 minutes.
There's a large parking lot outside the new terminal. You pay 5 yuan for the first hour and then 2 yuan per half hour. You pay 50 yuan per day. These rates apply to outside parking; the rates for the parking garage are somewhat higher.