There's actually no built-in way to import emails on your own to Gmail. But, there's a workaround—the same workaround to export your emails from most email accounts.
Just login to your new Google account and your old email account in a desktop email app like Outlook on a PC or Mail on Mac. Once your emails have synced, select the folders of emails that you want to move from your old account, and drag them over to your new G Suite or Gmail account. Leave the app for a while and let it sync; it can take a while to copy everything over if your email account has years worth of messages.
Once it's done syncing, open your new Gmail account, and you'll see all of your old emails organized with labels in lieu of your old folders. And you can search through all of your old messages or reply to restart an old conversation, just as you could in your old account.