Is there a train from Jinan to Dongying? and if so, how often does it run? and how much does it cost

Yes, there is only one train, 5011. The train leaves JN at 12:04 and reach Dongying at 16:20, in about 4 hours and 16 minutes.
The train's hard seat costs about 29.5RMB per person and soft seat costs about 46.5RMB per person. It is suggested to book in advance to make sure.

From Atlanta, you can fly to Shanghai or Beijing. Then transfer to JN city of Shandong Province. From JN, you can take a train to reach Dongying. This may be the cheapest method.