How to move the WordPress webpage to other hosting servers?

Crrently I am trying to moving my wordpress website to new paid hosting service godaddy. It is a wordpress managed hosting. I have gone throgh a lot on internet on moving wordpress from one host to other host. The suggested duplicator plugin was not working for me. Finally, I tried the follwing:-

Assume that my old website name is

www.old-site.org

My new (temperory) address is

www.new-site.org

I downloaded the entire website from www.old-site.org

Downloaded database backup from wordpress. Then edited the old wp-config.php replaced the database name, username and password with the new site’s database details. Finally used filezilla to move the old site files to www.new-site.org/new location. After all done, typed www.new-site.org/new/wp-login.php in firefox. A login window appeared and asked for the password. It looks like all done. However, after login the site addreas goes to www.old-site.org/wp-admin/. What is the mistake I have done and how do I overcome this issue? Do I have to replace URL in any file?

Edit:-

Ok. I did the following now. Downloaded the old *.sql database. Used http://pixelentity.com/wordpress-search-replace-domain/ lik for changing old url with new one. Then I gone to godaddy phpAdmin –> import database and imported the modified Sql file. But the result is same. After login the site is going back to old url.

Answer

1. Place all the files from public_html folder of old server to the new public_html folder.
2. Take the database backup from old server from php myadmin, go to import in it and import sql file.
3. open the sql database file in editor such as notepad++ or anything and find siteurl where you will find old website address as something like http://www.oldsite.com you have to replace all such old url in to your new url so use ctrl+f and replace www.oldsite.com to www.newsite.com and save the sql file
4. on your new server create new database and take note of its username, password and database name
5. on your new server open phpmyadmin and export the sql file to the new database so created.
6. open wp-config.php that will be in your public_html(root) folder in new server and replace database name, database user name and user name password (that you have noted in step 4). if followed correctly it should work.

Alternatively for database URL replacement you can use wp migrate db plugin that takes care of serialized arrays..

Also for complete automatic migration you can use all-in-one wp migration plugin which is also good