I am changing the DNS of a domain to point to a new server. What is the best way to handle this to make sure the transition goes as smoothly as possible, specially pertaining to email? Ideally I would like their current emails to not be lost. I believe they are just using webmail at the moment.
Thanks!
As long as the MX (mail exchange) records are not updated everywhere (could take many days), you can still get mail traffic to the old server at the same time when other mail goes directly to the new one. Configure the old server to forward all the mails it receives to the new server, either using numeric IP, or verifying that the old server has access to updated DNS records.
How you do mail forwarding depends on the mail software you run on the old mail server.
You didn't provide enough information to give you a great answer, but this may help:
It is non-trivial to move mailboxes between mail providers. If all of your users used local mail clients, then they could just download the mail locally and they wouldn't notice anything different when the DNS changed. If they are all using webmail, you'll need to run some scripts to move their mail to the new provider.
If you're worried about losing incoming email when you change mail servers -- don't! The sending mail servers will queue mail for redelivery. The only trick is that you need to clear out the old MX records with enough time for your TTL to expire before adding the new MX records.
T=0: remove MX records from DNS
T=1: set up user accounts and domain configuration at new mail server
T=2: add MX records for new mail server