Wordpress technically is written in PHP and has a MySQL (or MariaDB equivalent) database. If you are looking for hosting then you are looking for a company that provides those as options.
An alternative is where the Wordpress is configured and setup for you, and you don’t worry about the technical aspects behind the scenes. Wordpress.com is but one example of this. Think of it as Wordpress as a platform, or software as a service.
As for the content on your existing site, if you have an admin account then there is the option to export and import content . This means you can export from your existing site and import to another.
Wordpress has a system of plugins which modify or add functionality to the base Wordpress offering. Themes change how your site presents your content. Again if you have access to the admin account, then you will be able to see what plugins are being used, and what theme is in use. If they are public ones then you’d be able to install them on any new site you setup.
If the person who left didn’t add additional functions via a plug-in(s) but edited the Wordpress core files direct, you have an upgrade problem. Let hope they didn’t. Same with themes, hopefully no functionality embedded in current theme, which would be lost, should you change the theme.
A word on PHP versions. If you are moving from versions 5.X to 7.X then be prepared that some parts of your website may break. If you don’t have any plugins then that is the best situation for an easier move, and the functionality you’ve described is included in the base Wordpress offering.
For the domain name, you will need to find out who administers that, and arrange for it to point to your new hosting etc.
Generally you’ll move all your content, test it all looks good and the new site works, then point domain name at new host / site, then shut down old host .
