hello, im from germany, welcome to the world cup!
from dortmund to nuernberg, it will take about 4-5 hours by regular train...
from nuernberg to kaiserslautern (which is close to where i live) it will take about 2-3 hours by regular train...
if u take the ICE train (the superspeed train) it will take you approxiamately half the time... flying is possible from dortmund to nuernberg, that would take maybe 45 mins... there are companies that fly cheap in germany like easyjet or ryanair, amongst others. a early booked flight could be as cheap as 20-50 euros, which is even cheaper than train...
flying to kaiserslautern is not possible, as frankfurt is the closest airport and is 1,5 hours by train...
Sorry, but I have to disagree abou the flying thing, because I work in the airline industry. Air Berlin (a low coster indeed) offers flights Dortmind-Nuernberg right now. But: Those are feeder flights, because Nuernberg is their winter hub! In summer they will most likely go to direct flights again (Dortmund-Palma, Dortmund-Barcelona etc.) Such a hub in summer is very unlikely, thus no feeder flights. Ryanair and Easyjet don’t serve tis route at all.
There’s a small local airline, called LGW, who serve Dortmund-Nürnberg, but they operate Do228 (18 seater, turboprop) and a oneway is about EUR 200,00. So forget about flying!
I checked the trains again: 5:00 hours is the speed train from Dortmind to Nürnberg! Costs appr. EUR 100. Slower ones are between EUR 55 and EUR 75, but they run between 8 and 9 hours and you have to change trains 4 times....
You have to watch out for specials (group tickets, Interrail etc) The „Bahn“ just offered a championship pass for EUR 349, for the duration of the tournament, valid for all trains, as often as you like to...