We flew from the US with a direct flight to Tokyo Narita from IAD (Dulles) with United Airlines (it took approximately 15 hours). We then flew right away to Osaka (Itami airport, the small one of the two, and much more convenient to the city) with ANA. When you go to Japan from the US you basically skip a day but at least we managed to arrive early enough in the evening (of the next day!) to have a nice dinner in Osaka before collapsing in bed.
Tokyo has two major airports: the bigger Narita, which is also the furthest, and the more convenient Haneda. If you can chose, I would try to fly directly into the latter. On the way back we flew from Narita to Chicago and then Baltimore. We found our tickets on Google Flights ($1080 round trip).
For the trip to Amami Oshima we bought two separate one-way tickets. The first leg (Fukuoka-Amami) was with ANA, the second one (Amami-Nokyo Narita) was with Vanilla Air. Note that the two main domestic airlines JAL and ANA offer special fares to foreigners if you can prove that you are flying from outside Japan. With JAL it's called Japan Explorer Pass and with ANA is called Experience Japan Fare.
From and to Narita airport there is a very convenient and cheap bus (for 1000 yen, half of what the train costs) that leaves from Tokyo Station or Ginza and gets to the airport in less than 45 minutes.
The currency in Japan is yen, abbreviated in JP¥, and the conversion is roughly ¥1 =$0.01. So ¥100 is $1, ¥1,000 is $10, ¥10,000=$100. Which makes the conversion on the street pretty easy. One thing to keep in mind though is that Japan is much less common to pay with credit card (unless is a Suica or Pasmo card in Tokyo, for example) so it's very important to always have cash with you.
Usually people going to Japan choose the classic 10-day tour Tokyo-Kyoto-Nara-Osaka, with the option of adding either Shirakawa-go, Hiroshima or Nikko if you have time -and never expand it much more than that. Given that I personally think that 15 days is the bare minimum to explore a country that is so rich in culture and so full of wonders, but nonetheless that is so hospitable and easy to get adjusted to, I would recommend to visit also other areas other than Kānto and Kansai and possibly other islands too.
Japan is not "just the same" everywhere: every region, every town or village, park or small island has a little gem that deserves to be admired, whether it's a shrine, temple, garden or old historic house, or a small shop, a market or a bridge. They are all different and all dated from different times and eras and the influence that they could have had from other cultures (for example by the Chinese, or European ones in Kyushu) make them sometimes really unique. Often the true, authentic Japan experience is better enjoyed not only in the famous spots in the big cities but also in the corners of the country off the beaten path... so I really suggest to spend at least a couple of days outside the big cities, ideally in one of the many mishuku (small, family owned, bed and breakfast) that are everywhere, especially in the countryside. In the same way, nature from an area to the other can be very varied and diverse and there are many great parts of the coast, mountains or volcanoes and national parks that are usually very well organized and maintained. Japan is also extremely easy to travel by train, and once you get the discounted JR Pass you should really try to use it as much as possible (for the same price): it's really worth it!
These are the lesser known parts of Japan that in my opinion deserve a visit:
- Around Tokyo: Nikko, Hakone and Kamakura
- Himejii castle
- The art islands of Naoshima
- Hiroshima and Miyajima island
- Shikoku and Tokushima
- In Kyushu: Nagasaki and Unzen but also Kumamoto and Kagoshima
- In Onshu also the Kapanese Alps and Nagano, Kanazawa and the coast of Matsushima
- The sacred mountain of Koya and the Kii peninsula
- Okkaido
- The Ryuku islands (not only Okinawa and the ones around it) but also Amami and Yoron.
Choosing when to go to Japan is not secondary, because depending on the time of the year your trip may really be different. We went in June when it is not supposed too hot yet, but in the second part of our trip we caught the last tail of an unusually long rainy season (called tsu-yu in Japanese) which was really annoying in Tokyo -where we barely saw the sun in three weeks- but it also cooled off the temperatures that start to rise a lot around mid July. Surprisingly we had less heat in the South (Kyushu) than in Tokyo, but we were also in the middle of a typhoon in Amami (end of June) so if you want to see these islands I wouldn't recommend to go at the same time.