Should I use add_action(‘publish_post or add_filter(‘publish_post?

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publish_post doesn’t work for custom post types, the correct hook (action hook) is publish_{$custom_post_type}. You should use add_action() as this is an action hook.

I also tend to make use of the transition_post_status hook which is a much more universal hook as it fires everytime a post’s status is changed regardless. You can use $old_status and $new_status to check the previous and new status of a post then do something.

For a new post, you can something like this: (Requires PHP 5.3+)

add_action( 'transition_post_status', function ( $new_status, $old_status, $post )
{

    if( 'publish' == $new_status && 'publish' != $old_status && $post->post_type == 'my_post_type' ) {

        //DO SOMETHING IF NEW POST IN POST TYPE IS PUBLISHED

    }
}, 10, 3 );

EDIT

For post edits/updates, using transition_post_status, you can do

add_action( 'transition_post_status', function ( $new_status, $old_status, $post )
{

    if( 'publish' == $new_status && 'publish' == $old_status && $post->post_type == 'my_post_type' ) {

        //DO SOMETHING IF A POST IN POST TYPE IS EDITED

    }
}, 10, 3 );

For further reading regarding post status transition, you can check out the codex

EDIT 2

Custom fields, built in and from the advanced custom fields plugin uses the save_post hook, which runs after post transition, so trying to add custom fields won’t work on here

For custom post types, a new hook was introduced called save_post_{$post_type} to make things easier. This hook is also recommended by the ACF forum to update custom fields.

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