Problems sending mail: Sending outgoing mail via Gmail from my Web server--which I did, for example, when transmitting online contact forms--produced odd results, such as always including my own email address as the From address.
Buggy implementation in Mavericks Mail: Apple Mail in Mavericks treats Gmail differently than previous versions did; and on paper, that's a good thing. For example, it can now eliminate duplicate messages that previously wasted disk space and bandwidth. Unfortunately, Mail's performance with Gmail was so poor and buggy during my first week or so of using Mavericks that I wound up leaving Gmail open in a Web browser the whole time.
Some of my complaints are specific to Apple Mail, and I did consider switching email clients. But though I've tried many other clients (such as Bloop's $2 Airmail, the $10 Postbox, and Google's $10 Sparrow), I've never found one that offers all the features I rely on from a combination of Mail and half a dozen third-party plugins. In any case, even a perfect client wouldn't solve Gmail's privacy issues, outages, and wacky IMAP implementation.
How to move to another provider
Once I decided to drop Gmail, I had to settle on an IMAP provider to use instead. I investigated dozens of paid services, including FastMail, Hushmail, and Pobox. Although each provider had its strengths (some are especially good at spam filtering, while others have extra bells and whistles), I settled on a rather obscure choice: EasyDNS. I've long relied on this Canadian company to provide DNS services for several of my domains, and its $40-per-year DNSPro package, which I was already using, includes a full-featured webmail/IMAP service called EasyMail. Since my domains were already managed there, setup was extra easy--and switching to EasyMail Apart cost me no extra money.
The steps involved in moving your email from Gmail to another provider vary depending on the type and configuration of the server that the new provider uses, as well as on how you manage your DNS records. But in broad strokes, these are the main steps:
Start with a local copy of everything: I'm assuming that you use IMAP with Gmail, and that your email client is configured (as most are, by default) to store local copies of all your messages. You should also have a backup of all those messages (using the backup tool of your choice), just in case.
Change your DNS settings: If (and only if) you used a custom domain name with Gmail, you'll have to log in to your domain registrar's or DNS provider's site and change the records that currently point to Gmail's mail servers so that they instead point to your new IMAP provider. (Your new provider should be able to tell you what settings to use.)
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