Moving your site to another server is a relatively simple task with regards to the way in which your software works. In order to do this, you would follow the process below.
Step 1 - Your old hosting
Switch your site offline ready for transfer to your new server
Make a full backup of your database.
Make a full backup of your files and folders.
Step 2 - Your new hosting
Create a new database
Restore your backup from the above steps to the new database you have just created
Upload your files and folder, which you backed up from your old server
Ensure any folders which should be writable, remain writable on your new server. The folders which should be writable by the software are as follows
Step 3 - Software configuration
Open your conf_global.php file, which can be found at the root of your installations folder on the new server. In here you will need to edit the mysql username, password, and database name, to match the new database you have set up and restored to.
Log into your admin control panel and run through the support tools by visiting System>Support>Support, and selecting "something isnt working correctly". You will need to select next only until it states cache has been cleared.
If you have also changed the URL of your site, or the folder in which your site will be in (for example moved it from a subfolder into root), then you will also need to change the site URL within your conf_global.php file. You should do this prior to running the support tool in the steps above.
If you have a constants.php file (not all configurations will have this), then please ensure that you check any paths which may be present within this file as they will likely need to be updated.