Suggestions and advice

Hello,

My family and I (3 adults) are going to Japan for 6 days in July. We are flying into Osaka on the 10th and want to travel to Tokyo via bullet train on the 13th. Ideally we may use the local trains to get around Osaka and Tokyo during our time there.
What is the best way of travelling around, and with what kind of ticket?

Osaka, 10th to 13th
Tokyo, 13th to 16th

Any advice would be appreciated. Thank you