yes, backup the existing files and move to new server. test accounts and sites, make sure nothing is missing.
if using cpanel, transfer with that, it includes handy transfer tools.
once the sites are on your new machine, then simply update the DNS for those domains, keep the sites live on the old server for 48 hours to ensure people can still reach it.
if the sites use a database, then when you moved over the sites to the new server, they should be edited to the details of the new database server if its different to the one they used originally. then you should make a quick backup of the existing live databases, import to the new databases, and edit the current sites to use the new database.
Do this step as quickly as possible , and now for the next 48 hours, regardless of the server people get directed to, they should be able to see the same site etc, however if any material is uploaded to the site by users via the old server that content would subsequently get lost unless its in a specific location that you transfer over manually or something.
anyways, if you plan it out it wont be too much of a headache and at most you would probably end up with an hour of interuption
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You forgot to mention mail issues. Dealing with mail opens up an entirely different can of worms. You don't want mail delivered to both servers. You'll need to do the things mentioned previously, along with loweing the TTL in your DNS so that you don't have the long 24-48 hour wait. You'll also want to do things like forwarding mail from the old server to the new.