First time trip to Japan

We are a family of three ( 2 adults and our teeenager son that plan to visit Japan for first time on April 2018.
Our itinerary is planned as follows
April 1, 2018 Arrival at Narita Airport around 1.55 Local time
April 2 to April 6 Tokyo with side trips to Nikko and Kamakura
April 6 ( Take the Shinkansen tto Takayama. Overnight at Ryokan Asunaro
April 7 to April 10 Kyoto (with side trip to Nara
April 11 Kyoto- Hiroshima-Miyajima (overnight)
April 12 Miyajiam (Himeji and Himeji Castle) and take the latest night Shinkansen to Tokyo
April 13 Departure from Haneda.
As the time of our trip approaches, do you think that we should buy a 14 day pass or a 7 day pass would be fine If we activate our passes on April 6th to Takayama. What is the best way to move around Tokyo, Nara and Kamakura?
What is Suica?
Where can we hire portable wi-fi?
Thank you all in advance and wish happy travels.

Most of your questions are answered here:
https://www.japan-guide.com/e/e2361.html
There is also a pass calculator where to can find the out if it pays off.

Essentially transport in cities can be JR but is more likely to be bus, subway or private railways, so you can�ft use your jr pass nor does it pay off for short distance only but if you get one anyway and there is an inner city jr Line you can obviously use it.

Suica/Pasmo .... is explained here: https://www.japan-guide.com/e/e2359_003.html

And WiFi here: https://www.japan-guide.com/e/e2279.html

Enjoy your trip to Japan!

Your note about traveling around cities. I have found it easiest when in Tokyo to use trains & subway with an IC card. There is a loop bus route you can use to get round Nara if you don't want to walk. I have always walked when in Kamakura although buses are the answer there as well. I do walk a great deal when on holiday.

Your April 12 day is a big rush. Day at Miyajima 4-6 hrs. Then 6 hours travel with stop at Himeji. Add 2-4 hrs.

Your April 12 day is a big rush. Day at Miyajima 4-6 hrs. Then 6 hours travel with stop at Himeji. Add 2-4 hrs.

Himeji castle opens until 17:00 (you must enter before 16:00) and you might better reach Himeji by 15:00.

http://www.hyperdia.com/en/cgi/en/search.html?dep_node=MIYAJIMAGUCHI&a...

So you need to start Miyajima at around noon. Miyajima's main attraction is Miyajima shrine, Misen mountain and the aquarium. If you want to do all of this before noon it will be rather difficult. Most of Japanese tourist don't do Misen so if you climb Misen you will find more than half are foreign tourists :).

After Himeji it is rather easy to reach Tokyo on the same day: it will take about 3.5 hours.

In general putting Kansai area and Hiroshima/Miyajima into 7 days of JR pass period is rather difficult, but I think your itinerary is well coordinated.