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When I got to Japan, I am going to spend 8 days in Tokyo (doing central sight seeing), 5 days in Osaka (with 3 return train trips to Kyoto to see the area) and a day in Hiroshima then a train ride back to Tokyo.
My question is, will I benefit from purchasing a JR pass? Or will it be more cost effective to just pay for each train ride separately?
Short answer - the full JR pass for this trip, assuming a 14 day pass at 45100 yen, will cost pretty much the same as individual tickets and more specialised passes combined, excluding the pass' ease of use. I'd strongly recommend it, with maybe a couple of 1000 yen extra for subways and private railways you might come across.
Long answer -
Narita Airport Express return ticket + 'Suica' charge card - 5500 yen
8 days travel around Tokyo - 770 yen 'Tokunai' day pass + 230 yen contingency (for subway or outer Tokyo) - 1000 yen a day, 8000 yen total
Tokyo - Osaka - 8510 yen
Kansai (Osaka/Nara/Kyoto region pass) - 6000 yen for 4 days
Osaka - Hiroshima - 5500 yen
Hiroshima - Tokyo - 11340 yen
Total = 44,850 yen
Not much of a saving, given the additional hassle, but a saving nonetheless if you're on a tight budget.
Last thing - you can plan all sorts of trips via all means if transport in Japan at http://www.hyperdia.com/en/. Remember to exclude the 'Nozomi' and 'Mizuho' Shinkansen / bullet trains. They're not included.
