I am answering this so that there is an answer.
If you own a domain name you can optionally have email sent to that domain.
There are a great many options for this. The "Rolls Royce" approach is to buy email hosting from a third party such as Rackspace. The shoestring approach by many hosting providers or Domain name sales operations is to provide a redirection service. There is a lot of ground in the middle.
If the email is not bouncing then you likely have an email account set up. Your provider should provide help for POP3 or IMAP settings for use with a local client such as Thunderbird or Outlook. These are hosting questions. SMTP for outwards email can be problematic, use what works. You are not obliged to keep any service you sign up for, you can receive and send email independently.
Ideally you would have bought a package that includes some email rolled up. Ask if they have a "catchall" service.
A DNS enquiry for your domain name should reveal "MX" records.
Yes, it is possible to to have your Drupal server also be a Mail server, this is also not for beginners and may violate the terms of your hosting provider or your ISP (if this is on your own metal).
But really, the out-of-the-box solution is to use your personal ISP provided email and move on from there when you are ready or have sorted out setting up the new email account. Leave important traffic on this in case of outages. Otherwise these are "hosting issues" not Drupal issues.