Hi Eli and welcome to the Forum!
I really like this idea because you will have a lot of great possibilities and freedom,
Here are some general ideas that you can consider, also let me know if you have any specific interests and I may be able to give you some pointers for them too!
Fukuoka is down in the south of Japan while Tokyo is more in the middle/north in Japan, what you could do is slowly travel up slowly and make a number of stops on the way. We have a great map resource that can help planning the trip.
Since you arrive in Fukuoka (the trains station for Fukuoka is Hakata, it is an interesting story), I would recommend vising Kyushu first, this is the lowest main island of Japan. Personally I love this part of Japan but it may be a little hot during summer.
Some of the best places to visit include:
Buppu well knows for its Onsen and Sand-baths
Nagasaki it is a very beautiful town with a rich history and many interesting attractions
Kagoshima is a very beautiful city all the way down in Kyushu, in its bay is an active volcano called Sakurajima much like the Vesuvius in Italy. The beautiful view of Sakurajima and the clear Ocean is worth a visit alone. Recently the city was connected to the Shinkansen network and can now be reached quickly by bullet train. It also boasts a range of interesting historic sights.
I could probably write a book about Kyushu itself but I will keep it to a last recommendation for now, which is Mount Aso an other beautiful volcano in Kyushu that makes for a great day trips and houses many hiking trains. Also the train ride there on the [http://jprail.com/trains/sort-by-type/limited-express/trans-kyushu-limited-express-kyushu-odan-tokkyu-kumagawa.html] is well worth the ride.
Okay so on the mainland Japan, once you leave Kyushu the next big city you will come by is Hiroshima I would really recommend a 1-2 day visit here at least, the peace museum is something that will leave a deep impression and you can combine Hiroshima with a visit to Miyajima on of the most famous sights in Japan.
As you move up you could make a stop at Kyoto, Osaka or Kobe and enjoy the Kansai Area a little more before heading on. From Kyoto it is easy to travel to Kanazawa by express train. It is a very beautiful town, much like Kyoto but a lot less aimed at Tourism. It has a beautiful old Geisha Quarter, Castle and Kenroku-en Garden, which is one of Japanese 3 most beautiful gardens. After Kanazawa you can travel back to Kyoto or make a quick stop at Nagoya it is perhaps not the best city for tourism but makes for a great base to make day trips.
From Kyoto/Nagoya you can visit Takayama which is an old village in the mountainous mainland of Japan. The train ride is very beautiful and has many stunning sights as it follows a river all the way trough the mountains.
After all this travel you may just want to spend your last couple of days in Tokyo but if you want to travel more Yokohama and Kamakura make for great day trips.
Hope this gives you a better idea of the options available,