Hi,
I guess there are many good ways for migrating content, but if your data is not on Sharepoint now (but in classic file shares), you can use one of the following:
1. Map a network drive to SharePoint Online and drag-drop files and folders to the mapped drive and upload using that method. Reference: support.microsoft.com/.../2616712
2. Use SharePoint Workspace to achieve the same experience, with improved interface. Reference: office.microsoft.com/.../synchronize-sharepoint-content-with-sharepoint-workspace-HA101967275.aspx
Bulk uploading like this isn't optimal from the Search and indexing perspective, but it works.
Best regards,
Jesper Ståhle