Can you give an example of the custom permalink settings you are trying to use?
Also check the codex re pretty permalinks:
http://codex.wordpress.org/Using_Permalinks
Finally confirm a .htaccess file is being created in the root of your installation as this is required for custom permalinks.
Basically I just want to be able to select the option that has the post name appearing in the URL: Post name http://katesbeautyblog.com/sample-post/
But when I try to select that, none of my links work. Instead I am just redirected to my home page. The only option that works is: Default http://katesbeautyblog.com/?p=123
I will check out that link you provided – thank you!
With respect to your last point, I really have no idea – I am new to the blogging word! How do I set up a .htaccess file/see if it is being created?
FTP into your site (or use whatever file management application your host provides) & look for an .htaccess file in your root WordPress folder. Bear in mind that some ftp clients are configured to hide .htaccess files, so do ensure that your client is configured to show all files.
