JR Pass

Good morning,
We're travelling to Japan in Feb 2025 for 20 days. Our proposed itinerary is as follows:
Arrive Tokyo on 23 Feb (spend 2 nights) to look around
Travel to Nagano 25 Feb to 28 Feb to visit Snow Monkey and other sights (3 nights)
Travel from Nagano to Kyoto from 28 Feb to 5 Mar (5 nights)
Travel from Kyoto to Osaka 5 Mar to 8 Mar (3 nights) Visit Hiroshima and other sights
Travel from Osaka to Tokyo from 8 Mar to 15 Mar (7 nights) for sight seeing, go to Mt Fuji and others.

Are we saving money if we get the JR pass in advance or buy tickets as needed when we get there.

Also do you have any suggestions on places to see.
It's our first time in Japan.

Looking forward to your suggestions
Cheers,
Elizabeth and Michael