15/16 Days in Japan, JR Pass?

Hi,

My friends and I are planning on spending ~16 days in Japan.

Here is what our itinerary looks like this:
Day1- Arrive in Tokyo, go to Kyoto
Day2-Kyoto
Day3-Nara
Day4-Kyoto
Day5-Kyoto to Osaka
Day6-Osaka
Day7- Day trip: Himeji & Kobe
Day8- Day Trip: Takeda Castle
Day9- Osaka to Hakone
Day10- Hakone to Tokyo
Day11- Day trip: Mitaka
Day 12- 16: Tokyo

We are unsure whether purchasing the 14day JR pass will be worth? Or should we just purchase standard tickets between cities, get the single-day bus pass for Kyoto, and rechargeable cards for Tokyo and Osaka?

Thank you!

Try to plot in transportation cost from Hyperdia and see how it adds up. At a glance, day 2-6 won't be on the full utilization of JR Pass as you are travelling within Kyoto, Osaka and Nara only. I find JR lines are not always the fastest option to reach certain destination.

Know for sure: https://www.japan-guide.com/railpass/

Your itinerary includes long distance travel on Day1 and Day9 so if you want to take JR pass, it should be 14day version, 46390 yen.

If you fly from Haneda or Narita to Itami (Osaka), the cheapest fare is about 8000 yen and might be faster than taking trains, but this is depending on specific date and time and the airport.

https://www.ana.co.jp/en/us/promotions/share/experience_jp/
https://www.jal.co.jp/yokosojapan/

Note that JAL's Yokoto!Japan fare is applied only those who travel in/out with One World airlines. For both airlines, Narita to Osaka flights are few so you may not be able to find an appropriate flight.

As for your travel to Takeda and Himeji, JR West Kansai Wide Area pass might be useful.

http://www.westjr.co.jp/global/en/ticket/pass/kansai_wide/

There seems to be a common misconception
to think as if a total JR railway cost were always calculated by adding up fees by legs,
which can lead to a wrong conclusion that a rail pass pays off when actually it is too expensive.

When stations A, B and C are on a straight JR route in this order,
basic fares for A -- B -- C and for A -- B + B -- C
may be different in prices from each other.
Also it needs be considered how long time a basic-fare ticket is valid for.

To use a JR Shinkansen or Limited Express service,
you pay a surcharge for that service, in addition to a basic fare.

whether purchasing the 14day JR pass will be worth?

No;
a 14-day (Ordinary-Class) Japan Rail Pass (46390 yen)
seems not the best deal for this itinerary.

Your leg beyond Odawara for Hakone is not served by JR lines.
Depending on which sites in Hakone you visit,
a 2-day Hakone Freepass (4000 yen) may pay off.

The total amount of regular railway fees for the major legs
(i.e. to travel from Tokyo, around the listed places and back to Tokyo)
other than the Hakone leg
is estimated at less than 42000 yen per adult passenger.

When you plan
to have a flight to Osaka Itami [ITM]
with ANA Experience JAPAN Fare applied,
to use (5-day) JR-WEST Kansai WIDE Area Pass
for your trips in Kansai Region
(with your Nara trip made on Day 4 instead of Day 3),
and to take a Romancecar train from Odawara on the Odakyu lines,
the total amount of transportation fees for major legs
other than the Hakone leg
is estimated at less than 35000 yen per adult passenger.

My friends and I are planning on spending ~16 days in Japan.

When will you travel?
A domestic flight is recommendable only unless a stormy weather is expected.

Arrive in Tokyo

How is your flight scheduled?
Which airport will you fly into?
When you arrive at Narita Airport [NRT], you are not yet in Tokyo.

Day trip: Himeji & Kobe

That may be hectic; I would allow one day each.
What sites to visit do you have in mind?

Day Trip: Takeda Castle

Please note that trains are not very frequently available on the JR Bantan Line.
Takeda Station (of JR, in Asago City in Hyogo Prefecture)
is south next to Bantan Line's north terminus Wadayama Station.
A Limited Express train may skip Takeda Station.

Day trip: Mitaka

Do you plan to visit Ghibli Museum, Mitaka?
Then, a weekday would be better.
(A reservation is required to gain admission to the museum.)

- ANA: Experience JAPAN Fare
https://www.ana.co.jp/en/us/promotions/share/experience_jp/
- JR-WEST: Kansai WIDE Area Pass
http://www.westjr.co.jp/global/en/ticket/pass/kansai_wide/
- Odakyu Limited Express "Romancecar"
http://www.odakyu.jp/english/romancecar/

- Arukumachi KYOTO Route Planner
http://www.arukumachikyoto.jp/index.php?lang=en
- Kyoto City Bus & Kyoto Bus One-day Pass
https://www2.city.kyoto.lg.jp/kotsu/webguide/en/ticket/regular_1day_ca...
- Nara Kotsu: bus passes
http://www.narakotsu.co.jp/language/en/pass.html
- Japan Guide: Hakone Freepass
https://www.japan-guide.com/e/e2358_008.html