If you are changing hosting companies you do not need to change your domain name registrar also, if you don't want to. Domain and hosting are too completely separate things.

All you have to do is first sign up with your new hosting company, then ask them what their name servers are. These will be in the form of URLs, such as ns.hostingcompany.com

Go to the website of the registrar where your domain name is registered, and login with your username and password. You should be able to easily manage your domain account, and you can change the name servers to your new host. Every domain registrar is different, so ask them how to change the name servers if you have problems.

After you do this, it may take a day or two for the name to propogate, but your domain name should now "point" to your new hosting provider, and all you have to do is upload your website into the "public html" folder on your new host.