Pranish - a few thoughts on your itinerary.
You donft say which Tokyo airport (Narita or Haneda) you will be arriving at/departing from. Depending on the airport and where youfre staying in Tokyo, there are various ways of getting into town, though itfs usually the train or bus \ it just depends which airport and where to.
For Hakone, a lot of people do it as a day trip from Tokyo or as an overnight stay (or perhaps 2 nights). Nothing wrong with 3 nights but itfs more than most visitors would spend there. Also be aware that Hakone suffered some damage in the recent typhoon. In particular, one of the local rail bridges and some of the related line was damaged and is not operating. You can find information here -https://www.japan-guide.com/e/e5200.html
There are primarily two ways of getting to Hakone - either (a) the (JR) Shinkansen (from Tokyo, Shinagawa or Yokohama) to Odawars and then the local Odakyu train from there or (b) the Odakyu line from Shinjuku (including, for an additional cost, its so-called eRomance Carf). You may be interested in the Hakone Fee pass, which you can either purchase at Odawara or at Shinjuku, if coming from there.https://www.japan-guide.com/e/e2358_008.html
Whether or not a 7 day JR Pass will save you any money is hard to say - it depends somewhat on the above questions (which airport, where youfre staying etc.) and whether a 7 day Pass can be made to gfith your itinerary. If you do decide to get a Pass, you can find a list of agents here -http://www.japanrailpass.net/en/purchase.html#step_01
My advice would be to use an agent located in your home country, preferably in your city (if there is one). The 7 day JR Pass costs \29,650 so check the price (in your local currency) that any agent wants to charge you to make sure it is reasonable.
I would only get a JR Pass if it will save you money. Donft buy it for gconvenienceh. With a JR Pass, you can travel in an unreserved seat (if available) or you can make seat reservations for free. If youfre going to make a seat reservation (and we always have) itfs no more or less difficult to do that than it is to go and buy a ticket - so cost, not convenience, should be the deciding factor. It is certainly true that one of the (potentially) great benefits of a Pass is it lets you decide to go somewhere else on the spur of the moment. However, given you have only 8 or 9 days and youfre planning to spend those days in Tokyo, Hakone and Kyoto, I doubt that flexibility is of value to you, as you donft really have the time to go to other places.