Hi,
You can arrange it however it suits your preferences. The important things are to avoid backtracking as much as possible, and if you start to use your pass from June 1st then you need to be in Tokyo by June 14th. Personally I think that avoiding very long and tiring train trips is the best way. But you certainly can get your money's worth by riding from one end of Japan to the other. The pace may be fairly brisk though and since Japan is a very long country, you won't be able to see everything it offers by any means.
To maximize your benefits from the pass, you might spend your first 2 days seeing Tokyo, your 3rd seeing the Mt Fuji area, either Hakone or the Fuji 5 Lakes area (for either, the pass is of limited benefit, so use a regional pass like a Hakone Free Pass), then activate your pass for June 2-15, and then use the Narita Express with your pass on the 15th to leave Tokyo for the airport (assuming you're flying out of Narita).
From there you can spend a few days going up north and coming back, but the best of Japan is Tokyo and southwards.
Using the bullet train as much as possible will save time and increase your savings.
Some popular places to visit going north are NIkko, Dewa Sanzan, Mt. Zao crater lake, Yamadera, Matsushima Bay, Hiraizumi, Lake Towada and walking the Oirase Stream, and the Shimokita Peninsula. Do some research and see what suits your interests.
South of Tokyo, seeing Takasaki/Shirakawago world be a unique chance to see the old Japan, Kanazawa is popular and has Japan's most famous garden, Kyoto & Nara are the crown jewels of Japanese culture, and nearby Himeji is Japan's greatest castle. A few days in Kyoto are certainly not enough but since you are pressed for time, the biggest 3 sights there are the Kinkakuji Temple, Kiyomizudera Temple and the Fushimi Inari Shrine. Fushimi is the most time consuming; you could spend a couple of hours to over half a day if you want to romp over all the trails. And the Otagi Nenbutsuji in Arashiyama is one of the most humorous and whimsical places you could see in Japan - well worth visiting.
Spending a night at a Koyasan monastery can also be a memorable experience.
Moving southwards, Okayama has its famous castle, and one of Japan's top 3 gardens, Korakuen, one of Japan's Top 3, and also the Handayama Garden. If you'd like to get off the beaten trail for a while, the half day hike up to Konpirasan in Kotohira is stunning. Going to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
Hiroshima also has a number of good places. Miyajima is the most famous, and aside from Hiroshima itself, Onomichi with its temple tour and Kosanji Temple is spectacular. Just beyond Miyajima also lies Iwakuni with its historical bridge and castle on top of the mountain. Kurashiki near Okayama is also famous for its old canals. Beyond Hiroshima lies Yamaguchi, which has a long history, plus Japan's best old limestone cave, Akiyoshido.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely FREE!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
In terms of history and varied points of interest, Nagasaki is really the most interesting city on the island. It is worth a couple days to see the city, and another day to see areas nearby like Shimabara/Unzen or whatever interests you. Gunkanjima, while it takes 3 hours, will blow your mind.
That should fill up two weeks pretty fully. As you can see, you'd need a lot more time to see all that Japan has to offer.
Good luck.
