If you feel that it would be in your best interest to move your website to another host
, that is not impossible to do. You need for your site to be up and going and it helps if your web host is willing to release you. There are a few different steps that you can take to make this go smoothly.
Check with your web host about moving the files. Most hosts will be able to offer suggestions, usually requiring you to download the files using the FTP or SFTP method. You will need to have user name and password for your web hosting account. If you don't have this information handy, you can also contact your web host and have them reissue you this information. Make sure that if you are using a web designer that you are the one who is listed as the owner of the site, not them.
If your site is database driven, then you will need to download a copy. This will allow you to transfer your site to the new web host. Doing this can be a little tricky and if you are not familiar with the process, then it is wise that you seek out help. On some sites you can use a special utility called mysqldump, but you may also be able to create a backup file using only PHP. It is important that this task is performed correctly; if you don't know what you are doing it is best to have someone else take care of this step for you.
All of the data and stuff has to be finished and fully backed up before you tell your new web host that you are ready to move. If it isn't done then things could be lost. When you tell them you are ready they will access the current site and then they will transfer the domain name.
When they have your domain name transferred you are free to get your site all set up with your data to be uploaded. There is nothing left to do besides telling the former web host service that you will no longer be needing their services.