That's a big difference, with "inbox zero" requiring an email client with great archiving that works over multiple device types. Meanwhile, at the other end of the spectrum, the personal information managers need something more like Microsoft Outlook, with excellent search capabilities as well as a good storage contract on the hosting side because these types of inboxes are often tens of gigabytes (GB) per user.
Moving to another website consists of transferring the website’s files and databases, configuring your site with the new host, and directing your domain’s DNS to the new host. Once you pick a new site host, they can usually help you out with this process. The cost will depend on the host you’re switching to, but it can range anywhere from $150-$400.