First you can take airport shuttle bus to downtown for an hour which cost about RMB15, then switch city buses to where you want to visit. I must say it takes time, you’d better take taxi after you get into downtown.
There are a lot of nice places you must visit in Chengdu like Panda base or Leshan Grand Buddha. But since you have limit time here you can choose Kuanzhai ancient street of Qing Dynasty, Jinli pedestrian street or Opera and tea house in the city, for you will know how do native live here.
I am not sure what you are interested in and when you fly to here, I suggest that you could find a local guide who will offer vehicle, suitable tourist route, and language support, of course, you have to pay for that.