john is going to New York by the train tomorrow错在哪了？

is going to是现在进行时，表示正在做某事。而根据你给的句子，语境是将来进行时，所以错了。将is going to改成will go to就行了

正确的句子应该是 ：
John will be going to New York by train tomorrow.

by the train把the去掉。交通工具一般不用the,如乘汽车：by car,乘公交：by bus,乘飞机：by air等。

by train，不需要冠词