Traveling from Atlanta to Beijing air and from Beijing to Shanghai via train 4/3-4/9

You can't leave Beijing and the surrounding areas for 144-hour transit so going to Shanghai is a no-no. If caught by the government, you and your children will get a 5-year ban. Also you did not mention whether you'll be going to a third country and if you are not (I.e., going straight back to the US) you absolutely need a visa.

I am trying to avoid getting entry permit. can you make a suggestion on how to make a 6 day trip to china without a visas. I noticed that the 144 hour I could stay in the region near beijing. please give me a little more insight. worse case I will get entry permit but I would rather use that money and possibly go to another country I just dont know what country would be ok to travel to. Thanks for your response.

There are several ways to do this. You can buy one way tickets to Beijing, following by one way tickets to a third country or territory (e.g., Japan or Hong Kong) and another bunch of one-way tickets from that country back to the US. You also can buy two round-trip tickets, one from the US to Beijing and back, another from Beijing to a third country or region and back. However, you must take into account that sometimes getting visas may be a cheaper trade-off because separate ticket fees may be much higher than standard round-trip tickets to China. Also you need to watch for hidden stops in Mainland China which will render you ineligible for 72/144-hour transit as well as tickets that are not actually in transit in Mainland China.

Just get a visa, it will be much better for you. US passport can get a 10 year multi entry visa then you don't have to play the silly third country game. Then you are also free to move freely around the country.