Weebly site hosted with GoDaddy-Client wants to keep GoDaddy, but use Wordpress instead

I have minimal experience with Wordpress. I have a client who currently has a site on Weebly, with their domain hosted on GoDaddy.

What they want:

• They want to eliminate their Weebly site and use Wordpress instead.
• They want to keep their GoDaddy domain hosting, since they already pay for that.

Where I'm lost: Wordpress requires you to pay a monthly fee to have the "wordpress.com" removed from the Wordpress site domain name. So, will the client need to pay for that as well as the GoDaddy hosting?

Also, since I've never really started this process from scratch, what is the recommended order that I start this process? I'll need to rebuild the site manually, so I don't need a transfer service or anything.

I suggest you to purchase a good hosting with following requirement

1. PHP version 5.2.4 or greater.
2. MySQL version 5.0.15 or greater or any version of MariaDB.(Optional) (Required for Multisite)
3. Apache mod_rewrite module (for clean URIs known as Permalinks) And point your domian to that hosting server. You can purchase a hosting from GoDaddy also.