Without a good importing tool (warning: the GSuite importing tools are shit), I recommend "imapsync":

https://github.com/imapsync/imapsync

After cloning the repository and installing the listed dependencies (some Perl packages available in your Brew / Ubuntu repository already), you can do:

    imapsync \
        --host1 imap.old.com \
        --user1 "[email protected]" \
        --password1 xxxxxxxxxxxxx \
        --ssl1 \
        --host2 imap.new.com \
        --user2 "[email protected]" \
        --password2 yyyyyyyyyyyyy \
        --ssl2 \
        --errorsmax 1000000 \
        --syncinternaldates \
        --useheader 'Message-Id' \
        --noreleasecheck \
        --noexpunge \
        --automap 

Works great for importing to and from Gmail, but you might want to add a folder rewrite for the Sent folder (with --f1f2).

But with FastMail, for those interested, the out of the box importing tools in their web interface work great, so no need for any command line tools.

I've used imapsync in the past to migrate thousands of mail accounts from and old decrepit IMAP environment (with some weird edge cases) to a shiny new IMAP based mail platform.

It's a hugely flexible and versatile tool and it's quite zippy as well, can highly recommend.

Effort, not money. As a Firefox/gmail user who routinely shits on Google, it's mostly the lock-in of any email address that keeps me coming back. I've actually opened a few paid accounts, but effort to switch is real. I gave up partly through switching over the dozens (100s?) of services that send me emails. I would never get to the point where all of my acquaintances and relatives actually use the new address.

Why not setup email forwarding and just wait a few years for people to get the message?

This is precisely what I have done and it's worked great. All services I use (and care about) I've changed to use service specific aliases now and it works fantastically well. Effort wise I'd say I spent probably a couple of days migrating some ten odd years of email from gmail, setting up domains etc. All in all, the migration itself was done within a week. Only maintenance I do on this setup now is send receipts to my accountant, and update credit card details when needed.