Bamberg and Nuremberg are inside the same local public transport association, the VGN. Thus the VGN tariff applies which is not shown by the DB timetable. While you can do this trip on a Bayern Ticket, cheaper is to use a VGN TagesTicket Plus. A day ticket for up to 6 persons - albeit only 2 of them must be older than 17 years. You need the version covering the whole VGN area which is EUR 16,80.
- no past 9am restriction like with the Bayern Ticket on weekdays
- if bought on a Saturday this ticket is also good for Sunday
Seat reservations are on regional trains not possible at all.
And are also not compulsory in Germany with long distance trains either (exceptions: night trains and the few ICE Sprinter). Thus you can always just buy a ticket and go. With long distance trains not the cheapest option as for these you can get discounted specials in advance. The specials for regional trains (like the Bayern Ticket or the VGN tickets) are always available and can be (and are) best bought on spot.
Edited: 5:20 pm, March 01, 2013