from Frankfurt Airport
Start: FRA FrankfurtAirport
or from Frankfurt?
Start: Frankfurt(Main) Hbf
You can save if you travel on regional trains from Frankfurt to Bamberg. Select as means of transport "only local transport" for this. There are cheap specials if you travel on regional trains on Sa, Su or on weekdays past 9am.
From Bamberg to Kulmbach use a Bayern Ticket
Also for other possible trips from Bamberg. E.g. to Nuremberg or Bayreuth.