To cover the four places of attractions in 5 days, I suggest you such a route: Hanoi - Zhangjiajie National Forest Park - Zhengzhou (Shaolin Temple) – Beijing (Great Wall) – Chengdu (Panda Bear Zoo) - Hanoi. But I think it is very pressing to do it within 5 days.

From Hanoi to Zhangjiajie: There are 2 direct flights from Hanoi to Guangzhou daily, so you can fly to Guangzhou first and then transfer another flight to Zhangjiajie Airport.
Hanoi-Guangzhou: CZ372, 08:35-11:05, China Southern Airlines. 1.5 hours’ flight time.
Guangzhou-Zhangjiajie: FM9390, 13:40-14:55, Shanghai Airlines

From Zhangjiajie to Shaolin Temple, you can take the train No. K509. 14:26-05:30, 15hours. The train arrives at Zhengzhou and you can take a bus at the terminal (just opposite the railway station) to go to Shaolin Temple. It will take about 1.5 to 2 hours.

From Zhengzhou to Beijing: you can take a flight at the same day when return from Shaolin Temple to save time. CZ3179, 20:40-22:05 China Southern Airlines or CZ3173, 21:50-23:05. The you can have a one-day tour to the Great Wall the nest day.

From Beijing to Chengdu (Panda Bear Zoo): you can take a night flight at 20:00 or 21:00 when go back from The Great Wall. The flight time is less than 2 hour. The next day you can take a bus from Chengdu to the Panda Bear Zoo in Wolong. The bus from Chengdu to Wolong departs 06:40 every day and it takes 3 hours.

After that you can fly to Guangzhou from Chengdu and then return to Hanoi.