Fetch and store locally all your data before you even attempt to contact your old or new provider (Have a DB? Export the content)
If you are the registered owner of a domain and want it to move providers as well, make sure your registered email is correct!
Usually the new provider provides some change instructions (like a quitting form). Fill it out, send to the new. If you have a domain with your website you may receive a registrator code which you have to send to your new provider (to proof you're the owner). Choose hosting pakcet. And that's pretty much it (unless I forgot something)