NO to booking train tickets. Buy the Tagesticket plus day pass (about €20/day/2 persons) for unlimited travel in the Nuremberg area:
https://www.vgn.de/media/schienennetz-gesamtraum-vgn.pdf
https://www.vgn.de/en/tickets/all-day-ticket-plus/
(Note that you can use a ticket bought for Saturday on Sunday as well!)
It's valid to Rothenburg and to Bamberg - as long as you use the routes within the VGN zone.
I don't see the point of changing locations. You arrive in Nuremberg - the area's train hub - and staying there gives you more flexibility to go/see/do as you pleas. Nuremberg's old town zone is an absolute delight - and it may feel to you much like smaller old-world towns elsewhere.
https://munique-tours.de/private-tour-guide-munich-bavaria/Nuremberg-from-Albrecht-Duerer-House.jpg
https://edge.media.datahc.com/HI553412058.jpg
Bamberg is only 45 minutes away, Rothenburg only a few additional minutes by train - both easy day trips. Staying in Nuremberg means no luggage schlepping and the extra checking in and out. Nuremberg has a TON of things to see and do.