Metro in Osaka and Tokyo

Hello, I will spending about 15 days in Japan, from 6th April 2016 until 16th April. I will arrive in Osaka where I will spend the first 3 nights, then I will move to Kyoto for other 3 nights before going to Hakone for 1 night and Tokyo for 3 nights.
When staying in Osaka, my 2 friends and I will be travelling to Himeji, Kobe and Nara, before moving to Kyoto. I read that maybe the Kansai Pass will help me get on the trains to Kyoto, Himeji, Kobe and Nara; but what about the public transports (metro and buses) in Osaka and Tokyo? Will the Kansai Pass also be used for public transport? Do you think I'd better get a metro/bus card in Osaka and Tokyo?
Thank you very much for your kind help
Best regards
Marco

Which?
KANSAI THRU PASS (by non-JR groups) or JR's?

Read again both.
http://www.surutto.com/tickets/kansai_thru_english.html
https://www.westjr.co.jp/global/en/travel-information/

Tokyo is "Kanto" region so Kansai's pass not valid.
https://it.wikipedia.org/wiki/Regioni_del_Giappone

Hello, thank you very much for your kind reply, but after looking at the links you sent me, I still have a bit of confusion about the best PASS to buy. The KANSAI THRU PASS as far as I see cannot be used consequently for 2 o 3 days, this means that if I want to use it one day, I cannot use it the day after (The Pass can be used on unconsecutive days during its period of validity (example: Mon, Wed and Fri in the case of a 3-day ticket)

I write you here below my trip, will you be able to suggest me the best Pass?

6th April KANSAI AIRPORT to OSAKA (and then we need to travel on local buses/metro)
7th April OSAKA/HIMEJI and our return to Osaka we will stop at Kobe for a few hours, then back to Osaka
8th April OSAKA/NARA
9th April OSAKA/KYOTO
9-10-11 KYOTO around the town with buses
12th April KYOTO/HAKONE
13th April HAKONE/TOKYO
13-14-15-16 around Tokyo on buses and metro, and then back to Narita apt.

This is the trip we are going to do, what do you suggest me for all these trains, metro and subway?

Thank you very much in advance for your great help!
Best regards
Marco