18 day itinerary, which rail pass?

I was wondering if somebody could advise/ help planning our (we are 4 adults) trip to Japan in September, our current itinerary is outlined below but could be subject to change, we have accommodation booked in all places but the only one that is set in stone (i.e we would lose the deposit) is for Kyoto Day 8-13.

Itinerary

Day 1: Arrive Tokyo Narita (stay in Tokyo 2 nights)
Day 2: Tokyo
Day 3: Tokyo- travel to Hakone (Hakone free pass 3 days?) (stay in Hakone 2 nights)
Day 4: Hakone
Day 5: Hakone- Takayama (via Nagoya?) (stay in Takayama 3 nights)- activate japan 14 day pass.
Day 6: Takayama (with car rental) visiting Kanazawa
Day 7: Takayama (with car rental)
Day 8: Takayama- Kyoto (5 nights Kyoto)
Day 9: Kyoto
Day 10: Kyoto- day trip to Nara
Day 11: Kyoto- day trip to mt koya
Day 12: Kyoto
Day 13: Kyoto- Hiroshima (2 nights hiroshima)
Day 14: Hiroshima/ Miyajima Island
Day 15: Hiroshima- Tokyo via Himeji castle (3 nights Tokyo)
Day 16: Tokyo
Day 17: Tokyo
Day 18: Tokyo leave Narita airport


My main query is which is the most cost effective way of getting about, at the moment I am considering getting 14 day Japan rail passes activating on day 5, with a hakone free pass beforehand but it does seem like a huge expense for 4 adults, I have looked into regional passes but I can�ft find any combination that works out significantly cheaper does anyone have any advice on this?
We would also be open to shifting the itinerary a bit or cutting out Hiroshima altogether if it would be significantly cheaper.

We would like to hire a car for at least a couple of days just for the freedom, we were thinking of hiring in Takayama, would this be easy to do?
(also would be open to driving more if it would be more cost efficient than buying train tickets for 4 adults).

Does anybody have any advice on which rail pass/ train tickets to buy and about car hire?

Thanks!

Try this calculator: http://www.aoimirai.net/japan/jrpass_calculator.html

And yes, it is a big expense, but think about the price it would be if you just bought the tickets by themselves! You can look up the prices using Hyperdia (hyperdia.com/en) - it would be much more expensive without the passes. You are travelling big distances on a very fast train!

I think the 14 day pass will work for you, but only if you include Hiroshima & Miyajima. Using it for Days 5-18 works well too. You can use it on your last day for the Narita Express, and as for getting in to Tokyo on your first day, you could save money using the 1000 yen bus:
http://accessnarita.jp/en/home/
To start using your JR Pass from Hakone to Takayama, go to Mishima Station and then activate your pass. A Hakone Free Pass is a cost effective way to see the area.
http://www.odakyu.jp/english/deels/freepass/hakone/
Sometime in Days 13-15, if you can squeeze in Iwakuni, which is just a bit past Miyajima, it is well worth it. From Hiroshima to Shin-Iwakuni by bullet train it is just 15 minutes.
https://www.youtube.com/watch?v=0COOZXi66mM
For Miyajima, try to find the times of high and low tide, they give different perspectives on the Itsukushima Shrine, and at low tide you can walk on the beach to the giant torii gate.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-iCNZ6AdsKGUB-HPKNxP1...

If you wanted to squeeze even more value out of your pass and expand your experience, on Days 16 or 17 take a day trip up to Nikko. It's one of Japan's top sights.

Thankyou, will go for the 14 day pass!