From Beijing airport

Well, welcome to China!

From Beijing to Jining: You can take a train at Beijing West Railway Station to Jining directly. Taking airport shuttle bus line 7 and getting off at the terminal station, you can get to the train station. The train (2071/2074) to Jining departs from Beijing at about 22:43. It's cheap enough.
Meanwhile,since there is only one train for Jining, you can atry to get there by air quickly.

From Jining to Anyang: There is no direct train from Jining to Anyang, so you can take a long-distance bus at Jining Bus Station. The bus departs at 12:30 and costs CNY 99.

From Anyang to Changzhou: There are two direct trains scheduled from Anyang to Changzhou: K1101/K1104 and K233. They depart at around 16:23 and 16:10 respectively.

Good Luck!