Move wordpress site from different domain/hosting

I have been making some sites, and one of the things I like to do is make a temporary subdomain on my hosting account and add wordpress to that, and make the entire site. Then I can show the person I am making it for to make sure they like it before they spend money on a domain. I wish I could figure out how to do this smoothly because I like doing it like this

Is there a plugin where I could export a site/template after its done, out of wordpress to save to my computer so I can upload right to another domain with a different host?

If I am doing this concept wrong, what is a better way?

Thanks