Best&cheap transport for 14days itinerary

Hello all,
We will be going to Japan in Jan 2018 for two weeks. I have my destinations planned but unsure of the ways to get between my destinations. Can anyone help me with transportation (best and cheap ways) from one place to the next? Is it worth the JR pass?

Following is my rough itinerary:
9/1 Land in Tokyo (Haneda Airport) at 4pm. We are planning to stay in Shinjuku
10/1 Tokyo 
11/1 Tokyo 
12/1 Tokyo 
13/1 Tokyo
14/1 check out of Tokyo, head to Mount Fuji (overnight)
15/1 check out Mount Fuji, head to takayama check in and day trip to shirakawago
16/1 check out Takayama and head to Kyoto. (planning to stay near Kyoto station)
17/1 Kyoto
18/1 kyoto
19/1 Kyoto 
20/1 check out Kyoto, Head to Osaka (planning to stay in Namba)
21/1 Day trip to Nara from Osaka
22/1 Osaka 
23/1 Osaka fly to back home

Thank you for all your help :)

No best transports. Preferences vary with persons. Try to balance among time, price, and easiness. Moreover, you need to specify the exact locations of your lodgings. Also the departing time.

9 Jan T
Assuming you will stay at JR Shinjuku.
1704-1747, 43 min, 610 JPY, 2 transfers, HND-(Keikyu)-Shinagawa-(JR)-Osaki-(JR)-Shinjuku
1708-1755, 47, 710, 1, HND-(Monorail)-Hamamatsucho/Daimon-(Subway)-Shinjuku
1700-1755, 55, 1230, 0, HND-(Bus)-Shinjuku
1645 Sunset. Ground transportations permit to see night views of Tokyo. Monorail might be a fun. So, I prefer the below.
1708-1759, 51, 690, 1, HND-(Monorail)-Hamamatsucho-(JR)-Shinjuku

14 Jan S
Assuming starting from sunrise.
0651 Sunrise
0715-0900, 1h45m, 1750, 0, Shinjuku-(Bus)-Kawaguchiko
0708-1006, 2h58m, 2080, 3, Shinjuku-(Keio)-Kitano-(Keio)-Takao-(JR)-Otsuki-(Fujikyu)-Kawaguchiko
0707-1006, 2h59m, 2460, 2, Shinjuku-(JR)-NishHachioji-(JR)-Otsuki-(Fujikyu)-Kawaguchiko
The Bus is fast, cheap, and easy.

15 Jan M
0718-1505, 7h47m, 15610, 5, Kawaguchiko-(Fujikyu)-Otsuki-(JR)-Takao-(JR)-Hachioji-(JR)-ShinYokohama-(JR)-Nagoya-(JR)-Takayama
0811-1243, 4h32m, 5000, 0, Kawaguchiko-(Bus)-Takayama
Forget trains.

1350-1457, 67m, 2470, 0, Takayama-(Bus)-Shirakawago
1659 Sunset
1730-1835, 65m, 2470, 0, Shirakawago-(Bus)-Takayama
Not enough time to go back to Takayama before sunset. Allow the backtracking in the dark.

16 Jan T
0701 Sunrise
0700-1107, 4h7m, 4200, 0, Takayama-(Bus)-Kyoto
0800-1117, 3h17m, 9180, 1, Takayama-(JR)-Nagoya-(JR)-Kyoto
If you want to experience a Shinkansen train, take the latter. Or, just depart later to take a time to see Takayama itself.

20 Jan S
0703 Sunrise
0714-0800, 46m, 790, 1, Kyoto-(JR)-Osaka/Umeda-(Subway)-Namba
0714-0812, 58m, 920, 2, Kyoto-(JR)-Osaka-(JR)-Imamiya-(JR)-Namba
0743-0900, 1h17m, 800, 0, Kyoto-(Bus)-OCAT/Namba
I prefer the ground transportation, so select the Bus.

21 Jan S
0703 Sunrise
0706-0742, 36m, 560, 0, Namba-(Kintetsu)-Nara
0706-0742, 42m, 560, 0, Namba-(JR)-Nara
Nara has three areas; Nara Park, Toshodaiji, and Horyuji. Depending on your interests and pace, try to go along the triangle: Namba-(Kintetsu)-Nara (Park)/Toshodaiji/Horyuji-(JR)-Nara.

23 Jan T
0702 Sunrise
0705-0753, 48m, 1050, 0, Namba-(Bus)-KIX
0710-0756, 52m, 920, 0, Namba-(Nankai)-KIX
Nakai is cheaper.

Clearly, you have no chance to use a Japan Rail Pass.

I would consider taking the Nohi highway bus from Tokyo to Takayama, and then from Takayama to Kyoto. It will take an extra hour on each leg,but it is a direct bus. You will not have to transfer between a Shinkansen to a Limited Express train. The cost will be about half that of the trains. Here is a link:

https://www.nouhibus.co.jp/english/

The Nohi line would also be the bus line for the Shirakawa-go trip, so it is at least a good link for exploring that part of your trip.

Good Luck, in planning your vacation.