After arriving in Eger by train, how can we travel on to Baia Mare in
the Maramures province in Romania?
Trains departing from Eger to Baia Mare at 12:30 daily take 8.5 hours and two connections (at Füzesabony and Debrecen). This info comes from the DB schedule link at How to Look Up Train Schedules and Routes Online.
Three options:
Train from Eger to Debrecen and then Debrecen to Baia Mare. If you have time spend a night in Debrecen. Its worth the time to see.
Transfer Service. I imagine it would cost maybe 200 euro. But if you had a full car that isn't bad.
Guide. I know a top notch guide in Romania that would come get you.