Step 1 : download all your webbsite's Wordress files, export existing site database from PHPMyadmin Step 2 : edit downloaded database.sql with wordpad and use search and replace function, replace your old URL with New URL Step 3 : Login to your new hosting cpanel create a new database and user and Import database.sql through phpMyadmin which you have edited in last step Step 4 : Upload all of your WordPress files to new server Step 5 : edit wp-config.php and change database name, user and pass info with new Step 6 : Login to your new site's dashboard and goto permalinks settings and hit save/update button 2 to 3 times Start Browsing
There is another process which will make everything easier for you ... using SSH ... If SSH is active on both of your sites. You need not to download and then upload . Just by passing a command all the files from one server will be transfered to another server. If SSH is not active then you can ask your hosting company to enable SSH for you and this will be in a couple of minutes. Main problem is transferring of files. commands for wordpress transfer over SSH is easy .. You can PM me if you need further assistance
I would suggest that you read Wordpress' official documentation, you can find many ways on how to do it =) http://codex.wordpress.org/Moving_WordPress