Undoubtedly via the Shinkansen. It is usually expensive, but if you haven't yet arrived in Japan, buy a Japan rail pass as it will pay for itself even on one trip. The Shinkansen doesn't yet extend to Nagasaki so you'll have to use local trains to get to the nearest stop which is Fukuoka I think. You'll need to change to the Niigata Shinkansen in Tokyo. about 7 years ago
by air plane is better.
it takes 3 hours, include 1 transfer at osaka, cost jpy60000 by advance booking.
if you have rail-pass, taking shinkansen also okay.
but it takes 10 hours, you need transfer 2 times, at tokyo and hakata.
there's no direct bus service between the cities. if you prefer to choose the cheapest way, you should go to osaka and take a bus to nagasaki. about 7 years ago