Have you been follow my previously post steps to add your domain to Office 365? if not please check my previously post and do it. If you did and then it’s time to deploy Azure AD connect to sync your on-premises AD to Azure AD (Office 365).
You can download and install Azure AD Connect to your ADFS server or you also can install it at your domain server if you won’t use ADFS server for your infrastructure. you can install Azure Ad Connect with Express settings but I recommend install it with Customized settings, because you can adjust settings to match your requirements.
There is no more legacy public folder database, public folder mailboxes also knows as modern public folders, if you still need public folders, migrate them to Exchange 2016.
Public folder migration is a one-way process, if something goes wrong, you can roll back but any changes made since the migration will be lost.
This document serves as both a guideline for redeployment document for the Exchange Migration performed at the GDM Company. The process described mainly focuses on a typical transition of Exchange 2010 to Exchange 2016 environment, converting the Exchange 2010 server to Exchange 2016 Server MBX role. Additional role options such as the Edge Transport role, are out-of-scope within this document.