Moving to a new host should not be difficult; I moved from a VPS from one provider to a dedicated with another provider. First of all, you want to backup the Dolphin site files to a zip archive; the zip archive is going to make it easier to recreate the site. Don't create the zip with a root directory. You can also do a direct server to server transfer; which is how I moved my site, but that means you need root access to the server. If you are moving from a cPanel install to a cPanel install, I believe you can do a backup in the cPanel. However, personally I would just do a site backup.
You also do a full export of your database.
Now, once you have your GoDaddy account ready for use, you will upload the zip archive to where you wish your website to reside. GoDaddy will have a web root that may be different than the web root from your old host. Let's say that GoDaddy web root is /public_html, if you want your site to be in the web root; and it does not have to be but most running one site will just place the site in the web root, then upload the zip archive to the web root of /html. Once the zip has uploaded, and it may take a while if you have a lot of files stored, then go to the file manager in cPanel; it will probably ask where you want to go to, choose web root. Click on the zip archive and then use the extract, or expand, button on the file manager. Your website will be extracted to the directory. You can then delete the uploaded zip archive.
Now, you need to create a new database. Now, GoDaddy may have some restrictions in place on naming protocol; therefore, you may not be able to give the same name. Don't worry as you adjust this in the header include file. Once you have set up a database with a user and user password, go to phpMyAdmin and see if you can import the database export you created. If you get an error because the import file is too large, then you can use a simple php script to import the database file to the new database. I have had to do this on some servers when using phpMyAdmin and did not have access to the server. Others on here may have another solution; I can provide the php file to run if need be.
Once you have the files in place and the database in place, you need to edit the header include file; header.inc.php in /inc
You will need to change the database information as well as the directory root.
find $dir['root'] and update it to point to your new directory root; you can get this from your control panel.
Also, $db['host'] $db['user'] $db['passwd'] db['db'] need to be updated to the GoDaddy database credentials.
If GoDaddy is going to be handling the DNS servers, then they can adjust the DNS to point to your new site.