This depends on a number of things.
First, if you have a Control Panel account, where you have access to the server and can set up your own email and do other things, then there is a very good chance your new host can move your entire account to the new server.
If not, you can still use the Cpanel to copy/backup the site to your PC and then upload it to a new server. Be sure and get the database, if you do it this way.
If not, if you are just paying for "a website" very often the host will control that and you will wave goodbye if you leave. If you have FTP access, you can copy the site, but you will leave behind the database, which is where your time has gone, entering your goods.
There may be other considerations. From what little you have mentioned, the safest answer would be - It depends.