You can take Zhengzhou-Jinan 00:57- 10: 162 train. It takes 9 hours 16 minutes. Hard seat: 38 yuan, then go to Yantai. There are many cars and trains.
Train can choose 2582/2583 Jinan-Yantai11:13-17: 56 Express, 6 hours and 43 minutes hard seat: 38 yuan.
Besides, have a nice trip!