I am going to be rebuilding a new website for a new client. There current site is at a different host and registar. I want to transfer ownership. Which one do I do first? Domain name to new registar or website to new host?
Short answer? Both. Sorta.
Transfer the domain name first. Not for any particular reason except that it may take a while. I have seen these go in just a couple of hours and I have seen these go for a couple of weeks. It all depends on the registrars and whether they have their [redacted] together.
Having the domain transferred does not mean that you cannot still use the host for a period. So if your domain name is transferred quickly, at least you know you have that. And to me that is the scariest part.
Meanwhile, if the site is not user updated, set up a new host and begin transferring the site to a new host. Often that will not take too long depending on what makes the site go. You may need to do research if it is a blog, forum, store front, or whatever to know how to transfer the data. Hopefully it is a no-brainer. If it is a user updated site, then practice the transfer to ensure the install is okay. Then all you will need to do is transfer the site data again quickly at the last minute. You can set-up a sub-domain temporarily to test the new site.
If the site is static, then you can transfer the site anytime very quickly, but you still might want consider doing a practice run and test just in case.
Once the domain name is in your control, and assuming that the site has been transferred, all you need to do is to update the DNS to point to the new host.