The train only goes in one direction. Known stops of the train include the Human World as well as the world where the Mogeko Castle is located at.
Currently, it is unknown if there are other stops asides from the two mentioned above.
Role In Mogeko Castle
The train is the first location in Mogeko Castle, and Yonaka was riding it to return home. She decides to sleep on the train, and when she wakes up, she finds herself at a train stop called "Mogeko", where Mogeko Castle is located. Signs plastered around the station indicated that the train ran in only one direction, leaving Yonaka stranded.
Role in Carnival Rhythm
Evident in a few screenshots revealed on Carnival Rhythm's game page, it appears that the protagonist of Carnival Rhythm, Shrimp, is going to ride on the Mysterious Train.
The exterior of the train was revealed in some of the screenshots for Carnival Rhythm; the train appears to be silver on the outside, with a blue stripe on the sides of the carriages.