URL masking

I’ve read through the codecs and I’m trying to find out the best way to change or move my MySQL and subdirectory files to a new domain without extraneous effort. For instance, I built a stage site under my name/subdirectory/and have recently purchased a new domain for a friend and would like to migrate the data over to the new domain.

I spoke with my domain hosting company at length and makes like to me that it is not a hosting concern. That they would offer no support for anything WordPress. So I have everything working under a public_HTML/subdirectory… But I’m not sure what steps to take next. I’ve moved WordPress into different folders before but not in 2 different domains. Any help would be greatly appreciated because my host was completely unwilling.

Thank you in advance for your thoughts.