On my initial setup, I had installed CiviCRM onto the same database as the Wordpress installation. Now I want to split this as I am migrating the website and would rather backup and have a copy of the CiviCRM installation on its own. Does anyone have any advice on how this could be accomplished? Can I just backup the existing DB and restore to a new instance and it would all just work?
That's really two operations: splitting the combined db to separate WP and Civi ones, and migrating. For the migration part see the manual.
To split the db, you could make a duplicate of the original as the Civi db (dump and load), then in the Civi one delete all the non-civi tables and in the original/WP one, delete all the civi tables. In your
civicrm.settings.php file find the setting for
CIVICRM_DSN and change it to use the new database name.
Make sure you keep a backup of your original combined db just in case!