I know that we don't need to use prepare($query) when we put it in insert() and update() because these functions sanitize data for us but I'm not sure if we need to prepare query before putting it in get_results(), get_row() and get_var()?
Thank you.
If your query has no dynamic input, then you technically do not need to use $wpdb->prepare(), however that assumes your own input is safe.
If your query contains parameters in which are dynamic, especially supplied by users from a form as an example, then you will want to use $wpdb->prepare().
Using $wpdb->prepare() is just good practice, but in order to use it, you must pass at least two arguments (the query, and at least one placeholder).
Please note: As of 3.5, wpdb::prepare() enforces a minimum of 2 arguments.
See: https://make.wordpress.org/core/2012/12/12/php-warning-missing-argument-2-for-wpdb-prepare/
prepare() internally, like insert() and update() do... It's up to you to sanitize data for methods in question.
get_results() but that's what prepare() was intended for, doing the sanitization for you. I recommend reading the source for prepare() in wp-includes/wp-db.php around line 1210 to see exactly what this method does under the hood for you.