Itinerary advice

Hi. First time posting, first time planning trip to Japan with my wife (both early 50s but reasonably fit), and incredibly excited- seeing Japan has been a dream for as long as I can remember.

Any advice/opinions/suggestions on my first attempt at an itinerary for 13 x nights mid-April 2020 would be most appreciated. Main interests would be to see the sights, experience the culture, enjoy the food and relax for a bit.

Arrive Narita Mon 6th April 2.00pm, head to Tokyo, stay in Shinjuku area hotel 4 x nights, giving 3 days exploring/day trip to Mt. Fuji.
Fri 10th April travel to Kyoto, stay in Downtown hotel for 4 x nights, giving 3-4 days exploring/day trip to Nara.
Tues 14th April travel to Takayama, stay 1 x night in hotel, 2 x nights in Onsen area Ryokan, then back to hotel for 1 x night. Or alternatively Hiroshima + Miyajima?
Sat 18th April travel back to Tokyo for 1 x night then up to Narita on Sunday to catch 11.30am flight home.

Once again...any thoughts and suggestions very much appreciated.

Stephen

Sounds to be a great plan for a first trip.

Hiroshima and Miyajima are nice but personally I would save them for a later trip and go out to Takayama. That way you will see a completely different side to the country, with spectacular scenery through the trip especially if you take the wideview hida there.

Try to squeeze in a stop at Hikone (near Kyoto) or Matsumoto (closer to Takayama scheduling wise) for a look at a great original castle.

Your listed interests are extremely vague. What does "see the sights" mean, exactly? You need to narrow your focus according to your interests - there are many aspects to choose from - arts, temples/shrines, castles, nature, hot springs, hiking, festivals, shopping, museums, gardens, folk crafts, pop culture, cuisine, and many more.
There is no one size fits all.
Have you looked through this site with its listing of suggested places to see? That is a beginning. Or try the JNTO page at https://www.jnto.go.jp
Some regional info is also helpful, such as for Tokyo:
http://www.gotokyo.org/en/index.html
https://goo.gl/qyQtGV
https://tokyotouristinfo.com/en/
And for Kyoto:
http://www.kyoto.travel/
https://goo.gl/knQtMg
And adding Nara is definitely very worthwhile as well.
http://narashikanko.or.jp/en/
https://goo.gl/LfvWnL

In addition, going to the tourist info offices at the main train stations can be very beneficial.
And if you need to choose between Takayama and Hiroshima/Miyajima, I'd pick HIroshima easily. Lots of great and varied sights, and easy to get around.
https://goo.gl/fAz4ax