We’re needing to move all of our existing content from one Office 365 Tenant to another, including Email. Does anyone know the faster and cheapest way to do this?
Please refer these articles to get some help while migrating Office 365 tenant manually:
For automate migration, you can also take a look at Lepide Exchange Migrator. Here is an article http://www.techieshelp.com/using-lepidemigrator-for-exchange/ describing how it helps you in tenant migration.
For the SharePoint Online sites, you may re-create the site collections in the new tenant. Later, you may save the previous sites as templates (including the site content), download them to your local PC and then upload to the SharePoint Online of your new tenant. See Migrate SharePoint Online to different SharePoint Online Office 365 Tenants
Thanks,