First time to Japan

Hello,
Trying to organize my first trip to Japan for next month. So my idea is Tokyo for about 4 days. Might include a day trip to Nikko or Hakone. Then heading to Kyoto for another 4 days ( maybe 2 days in Kyoto and 2 days in Airashiyama) with day trips to Nara o Hiroshima/Miyajima.
Then back to Tokyo, since I'll be flying to Taipei.

What's the best way to move around? (I'll be traveling with my mother who is an active 70 y old woman) From airport to hotels?
Within the cities? From one place to the other? Is JRP the best option? Should I buy it in advance? Should day trips be booked in advance as well?

Looking forward to hearing suggestions!

Train will be the best way to get around.

If you've only got 4 days in Kyoto, and want to spend 2 in the city and 2 in Arashiyama, how will you have time for day trips to Nara and Hiroshima? Arashiyama should only take a day, depending on what you want to see.

What airport are you flying in/out of? There are two that serve Tokyo. There is information for how to get to/from both on this website (use the search function).

Likewise, there are "Getting to/around" pages for each location on this website, that describe the various travel options.

A 7-day JR Pass will pay off if you travel to Hiroshima. Just Tokyo-Kyoto-Tokyo will be just short of the cost of a JR Pass, but you could include one airport trip in there.

It depends on where you are staying (which hotel). It helps if your hotel is near a major train/subway station.

From the airport to Tokyo, there are trains and buses. Which train or bus to take depends on where your hotel is.

From one city to another, there are express trains (e.g., Shinkansen) and express buses.

Within the city, there are trains, subways, buses, and monorail.

For visiting Arashiyama, there is no need to change your Kyoto accommodation. Arashiyama is in Kyoto.

A day trip from Kyoto to Hiroshima is doable, but it will be tough. I recommend to stay in Hiroshima 1 or 2 nights.

Have you booked your flights already? It would be easier to fly from KIX to Taipei.

I agree that Arashiyama can be seen in one day, spend another in Higashiyama and a third in the north or the south part of Kyoto.
Why bother with a day trip to Hiroshima when there are many places closer to Kyoto you can see. Himeji and Nara are the obvious ones but there are others.
Nikko will take 2 hours to get to by train and I think that it deserves more than a day trip, especially if you are going in late October/early November when the autumn colour will still be there.
If you've not been to Japan before consider doing fewer places in depth rather than more places inadequately. You can't get a good understanding of Tokyo without 4 days there I would say and Kyoto needs 2/3 days too.
Regards, Richard