At some point you will want to consider moving your inbound mail routing from your on-premise mail system to Office 365's ForeFront Online Protection for Exchange (FOPE) If you do not have any on-premise gateway mail routing requirements, you may want to consider this move when most of your mailboxes are in Office 365. If you plan to do a flash cutover, then you may want to consider the MX change right away.
Moving your MX record does not mean on-premise mailboxes will stop receiving e-mails, this is why the Hybrid option is available for Exchange on-premise organizations. You may have other on-premise mail routing requirements, which should be evaluated before changing your MX record.
To get started, you first need to set up your Office 365 FOPE settings to receive e-mails. Go to the Office 365 Admin Portal at https://portal.microsoftonline.com. Click on Manage under Exchange Online: