Current price in JPY is ¥29110 for a standard 7-day pass. Anyway it is cheaper than 2 individual long-distance shinkansen trips.
The Japan Rail Pass is 28,300 for 7 days. If you are going round trip to Tokyo-Kansai within a week it is more than worth it. If you are going one way (or will travel more than 7 days apart) it probably isn’t.
JR will take you from Tokyo to Hakone, Hakone/Tokyo to Kyoto, Kyoto to Nara, Nara to Osaka, Kyoto to Osaka, and Osaka to Tokyo. Within Kyoto, within Osaka, and within Tokyo you will likely have to pay for local subway or buses (and will incur those costs anyway).
In addition to the nationwide JR Pass, there are several area-only passes which offer unlimited train travel within a specific region, for example Kanto (Tokyo) or Kansai (Osaka/Kobe/Kyoto). There is a nice list with restrictions and prices on Japan Guide.
As Michael said, you still need to do the math to see if buying the pass is going to be worth it but there are many options available which might save you some money depending on your travel plans.
If moving long distances inside Japan as cheaply as possible is your prime concern, then highway buses might be a good option. Check out the highway bus page on Japan Guide for routes and operators.
This page lists what lines the JR pass is valid for – it includes buses operated by JR, but that’s not all of them, especially not in cities. More importantly, the JR pass is not valid for subway lines in any of the cities you mentioned. In Tokyo at least it can be used for some metropolitan JR lines, most importantly the Yamanote circle line.
The rule of thumb is: look up what your long-distance train fares would be if bought directly, and buy a JR pass only if it’s cheaper than those. Local usage is only a small bonus.
I’m not aware of a way for tourists to save much money on local transportation. Renting a bicycle may be an option. You may also want to get a Suica Card, just for convenience.
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