"Go into the train" for me insinuates brutally colliding with it. However, "I went in a train" (especially said by someone possibly excited about the new experience of being inside a carriage) sounds correct, but "to go by train" or "to go on a train" are in standard use.
EDIT: Sorry I think I misread your question. If the train is already in the station, one would "go into the train". This seems perfectly normal English to me.
EDIT: and I think "get in the train" would be the most common verb form using "in".