Walk through the steps!
Here are the steps taken to help you transfer your website from your existing provider to GreenGeeks without downtime.
Once GG team has notified that your website transfer is completed, they'll ask you to double check your website to ensure that all is working properly. Provided that you changed the DNS and waited the 24-48 hours for the domain to resolve, everything should be fine.
The best thing you can do is keeping the old hosting account active for at least one week. This way if there were any issues during the website transfer process, it would be possible to correct it easily. Once you're confident that your website is working smoothly, you can proceed to cancel your web hosting account.