Get Posts that are in the current month or later

You can use meta_query, if you date information is saved in custom field. Like this:

$args = array(
    'post_type' => $custom_post_type,
    'posts_per_page' => -1,
    'meta_query' => array(
        array(
            'key' => 'event_date_and_time',
            'value' => current_time('Ymd'),
            'compare' => '>='
        )
    ),
    'orderby' => 'meta_value_num',
    'order' => 'ASC'
);

This code suppose you date is saved in YYYYMMDD format (20150820). And it shows post from today to future, so you will have to change value to “first day of current month”.