Hi,
Well, 2 weeks is too short to cover nearly the whole length of the country, so you can see some of the best known sights and move on. You certainly can get your money's worth by riding the rails so far though. You might start with Sapporo and do a day trip to Otaru, or if you want to see Hokkaido at its best, in July the lavender fields in Furano and Biei are in full bloom. Then move on down to Hakodate. After there you can see Aomori: the Shimokita Peninsula if you like, then go to Lake Towada and walk down a bit by the Oirase River which is quite famous. Some other popular places in the region are Hiraizumi, Sendai/Matsushima Coast, Yamadera, Zao, Dewa Sanzan, then proceed to Niigata and Tokyo.
If you have not seen Nikko, Kamakura and Hakone, they all make great day trips from Tokyo. From Tokyo, you can proceed down to the Kansai Area (Kyoto/Osaka/Nara/Kobe) or if you want, to can go see Takayama and Kanazawa first. Definitely see Kyoto and Nara though, then you can move down to Okayama and see the castle, plus Korakuen and the Handayama Gardens. Further south is Hiroshima/Miyajima, then Iwakuni for its iconic bridge and mountaintop castle.
Not to be missed in the area is Akiyoshido which is a 300 million year old limestone cave 1km deep with an underground river and giant caverns. Beyond Yamaguchi is Kyushu and Fukuoka. You might want to spend a few hours in Kitakyushu to see Kokura Castle and the wonderful night view from atop Sarakurayama.
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.
Actually if you could go visit Nagasaki for at least a day, it'd be well worth it too. It's really the most interesting city on Kyushu.
Most likely there is too much for 14 days listed here. You should go through the various places and see which you want to see the most, and then flesh out your itinerary.
Regarding taking the ferry or hydrofoil to Korea though - you will need some paper showing a ticket or reservation, otherwise you will likely be denied boarding at your airport departure - unless you have a visa to stay in Japan, you need to have a reservation out of the country within the allowed period of your landing permit.