If your host can simply point your new domain to your wordpress install:
Login to WP. Clear server cache, in fact deactivate for this step if in use and especially any CDN. Change site URL in settings. Logout. Clear browser cache. Change pointer at host. Once they have confirmed, login. Note, make sure new domain was setup (for you) at least 72 hours previously. Done.
As for SEO, make sure to include 301 redirect rules in .htaccess immediately after all steps above are done. Make sure any SEO related plugins/settings are updated right away also...there will be some affect on SEO. Make sure any webmaster tools/analytics are done right away too..(all pointing to new domain)
Many think it's the domain name that matters - not as much as site content and doing things right...why the change then?