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Dear Ms. Fogelman.
Thank you for posting your request on History Hub!
During that time frame, using the train was a very popular method of transportation for immigrants entering the U.S. at New York and travelling westward. Carriages and wagons were still in heavy use during this time. The Erie Canal also would have been a possibility to take part of the trip via boat. Further, there were early automobiles available (though you have to consider if the family would have had the means to afford the car and gas); therefore, it is possible (and there were roads to support) that a large portion of the trip could occur by car.
We hope this is helpful. Best of luck with your research!
[Information provided by Elizabeth Burnes, Subject Matter Expert]