Moving A Site - A few basic, but perhaps over-complicated, questions...

We are setting up our new site on a different server (Server B) and dev address (Domain B) than our current site and url (Server A & Domain A.) When we're ready, we'll be transferring Domain A to the new hosting environment on Server B.

I think that technically, we're not moving the site/files but we will need to change the WP setup so that all of the work we've done will be seen at Domain A rather than Domain B (our dev address.)

Is there an easy way to do this or does anyone know of any guides that we can follow to easily move a site from one domain to another if they're hosted in the same environment?