Should be just a matter of setting the sites up on the new server, FTP files, set up databases, to match the old sites, then once you've thoroughly tested everything in the new spaces and are sure everything's working, change the DNS server settings with your registrar, leaving the old sites in place for at least a week just to be on the safe side. DNS propagation can take anywhere from several hours to 3 days, maybe longer if there's any problem, past 3 days I would consider a problem and start looking into it.
Propagation of the domain name to new namme servers is variable time-wise depending on where you are on the worlkd wide interweb network thingie. Some guy in Oklahoma may see your site on the new server in a matter of hours while someone else in Michigan, New York, Guadalajara etc may see it a day, or 2 or 3 later. Name servers are maintained all over the internet that routes traffic too and fro, some owners of these routers update frequently, others less frequently, hence the variability.
As for advance setup and testing using Dremahost, I can't really speak with any comfort level about their dedicated service but assuming it's the same as shared (and it should be awfully close if you took their default managed config), you would just go into the web panel and add the domains. You might want to give some thought first about USERS however. Each USER has their own default directory/space on the server, and when you "add a domain" through the panel you will have to specifiy what user space it belongs in. I say you should give the users some thought first because you could put all domains (similar to html_public dirs on other *nix systems) under the same user's space if you're only goin to have one user maintaining them, or you could set up an individual for every website property/client/domain (whatever the case may be) so that you could seperate things out for a more flexible system in the end. If you did a user for every client/domain then you could allow some level of access if need be later without having to give them the single user/pass that would provide access to everything.
Once you add the users and domains, you might need to give their system a while to create everything, user spaces, domains in those spaces, authorization, etc. This may be instantaneous with a deidcated serrver, I don't know.
Once all the users and domains are accessible via FTP you can go back to the panel and set up all the databases you need. Again, not sure how this will be handled for dedicated servers becasue their shared systems use a cluster of database server, you should have your own. Instead of ramble on about how it's done with their shared system, I'd just advice you to contact support about this. If they say it's the same setup as their shared system, in that you need to setup a hostname, database, user and pass, come back and say so and I can probably walk you through that stuff.
Okay, so now you have all your http and dbs stuffs set up and waiting on files/data. Do your files transferring (ftp or whatever) and DB importing (shell or phpmyadmin {probably preinstalled somewhere, ask support}).
Now that everything's setup, you'll need to test using a web browser, and here is where I'm befuddled. Since your DNS would still be pointed at the old service at this point in the migration, you'll need to find a way to test these. I think you might be able to use directory mapping but surely there's a better solution than that, again, ask support about this one, you might even refer them to this post to see if they think my advice is sound, or point out where it's flawed.
Once you've figured how to test, test, tweak, and are sure all the web stuff is "ready", then you'd wanna be considering email. This could get ugly
You'll need to consider all your client's current mail setups and try to duplicate them on the Dreamhost servers so you'd have to do as little mail client tweaking for your cusomters as possible. If you're able to get all the mailboxes setup using the same user names and passwords and mail servers (mail.thedomain.com) then you're a lucky man and again, it should just be a matter of propagation, their clients will be checking mail as normal and if all the details are the same, once the DNS changes propagate to their location, the traffic will just be seemlessly routed from the new server.
Big job