That's quite a lot to pack into a week. I'd cut Nagasaki for sure - it's just too far if you're planning on taking in all the others plus
flying in and out of Tokyo. Really, Tokyo and Kyoto alone can fill a week, especially if you stick a day-trip or two in there.
Regarding Mt. Fuji, are you actually wanting to stop
there or just see it? Weather depending, you can catch a good view of it along the bullet train west from Tokyo (to Nagoya/Kyoto/Osaka/etc.). But of course, if you're wanting to savor it for a longer spell, you'll need to stop somewhere en route. One of the most popular spots to see it in all its picturesque glory is Hakone, which is an easy day-trip from Tokyo. The other option is to check it out from the other side, such as at one of the Fuji Five Lakes in Yamanashi Prefecture - Kawaguchi-ko is the largest and most easily accessible (by bus . . . the train's a hassle). But if you're wanting to climb the mountain, you're coming at the wrong time: the official climbing season is from July 1st to August 30th.
Out of curiosity, why Nagoya? As a current resident of the city, I have to admit that it's not a big tourist draw. It's definitely gotten more cosmopolitan over the years and it's a nice place to live, but without a specific reason to come here, first-time visitors don't usually bother. Apart from a modern ferro-concrete castle replica, a few museums and the rather staid Atsuta Shrine, there's not a whole lot to see. The one major advantage of the city for tourists is its usefulness as a base - places like the Kiso Valley, Takayama, Ise, Inuyama and Matsumoto are really easy to hop over to from here. But with your tight schedule, that's not much of an option.
Hiroshima can be done from Kyoto: it's only about an hour and 45 minutes away by bullet train. That said, it'd be a shame to go all the way out there and not see Miyajima; the pair are best done as an overnighter, with one day given to each. You could try and cram both into one day, but it'd be exhausting. I do highly recommend both of them though - they're among Japan's highlights.
As for the train. . .
If you're coming for a week with plans to start and finish in Tokyo, as well as include Kyoto, Hiroshima and potentially Nagoya, you should definitely
get a Japan Rail Pass. A one-week pass is ¥28,300 for each of you (don't get the Green Car version - it isn't really worth it), which is considerably less than all those individual tickets combined. But you must
purchase this before you leave the country. For specific information, visit the Japan Rail page
Don't worry too much about different rail networks. Yes, it's true that they exist and that the JR Pass won't cover you for them, but you're only likely to have to use them if you travel to certain locations in the metropolitan areas of Tokyo, Osaka or Nagoya. Of the destinations previously discussed, the only place where you'd need to go off the JR network would be for Hakone.