I can speak to what I did, and I recommend people at least start doing this immediately, even if they don't intend to switch email providers. Buy a domain name of your own, make an email address at it, and forward it to your current email provider. Every time you log into a website, make sure to update your email address to your new one.
Now, you're no longer tied to one company! When I decided to switch to FastMail, eighteen months later, I repointed the domain, and the vast majority of my email started flowing to the new service automatically! As a side bonus, if your email provider ever decides to ban your account (as Google has been known to do for spurious reasons), you can repoint your domain somewhere else, and you won't have to worry about losing access to all your other accounts and online assets.
You can use an IMAP email client to transfer your old mail, of course, and then I occasionally check my old mail account for stragglers. I chose not to auto-forward it so I'd clearly see what was or wasn't routing through my old Gmail account. I used to check it as often as I check FastMail. Then every couple days. Now I check my old Gmail account every week or two.