hi @regentcid . It's my first time travelling to Japan and hope I could get the best guide from you.

I've booked flight and AirBnB for my trip that falls on 16 May 2016 to 24 May 2016.

My main concern now is that I'll be around Osaka from 16th May to 19th May. And we plan to go to Tokyo on 19th May yet come back Osaka on 23rd May to catch our flight.
To simplify it.:-
16/5/2016 - 19/5/2016 Osaka
19/5/2016 - 23/5/2016 Tokyo
23/5/2016 - 24/5/2016 Back to Osaka

May I know by travelling like this, which method would u recommend for transportation? Should I purchase JR pass or buy it in Japan instead?
Which is the cheapest way to travel 2 ways between Osaka and Tokyo?

Thank you in advance!