3 Steps for Creating AJAX Post Filters

<form action="<?php echo site_url() ?>/wp-admin/admin-ajax.php" method="POST" id="filter">
	<?php
		if( $terms = get_terms( 'category', 'orderby=name' ) ) : // to make it simple I use default categories
			echo '<select name="categoryfilter"><option>Select category...</option>';
			foreach ( $terms as $term ) :
				echo '<option value="' . $term->term_id . '">' . $term->name . '</option>'; // ID of the category as the value of an option
			endforeach;
			echo '</select>';
		endif;
	?>
	<input type="text" name="price_min" placeholder="Min price" />
	<input type="text" name="price_max" placeholder="Max price" />
	<label>
		<input type="radio" name="date" value="ASC" /> Date: Ascending
	</label>
	<label>
		<input type="radio" name="date" value="DESC" selected="selected" /> Date: Descending
	</label>
	<label>
		<input type="checkbox" name="featured_image" /> Only posts with featured image
	</label>
	<button>Apply filter</button>
	<input type="hidden" name="action" value="myfilter">
</form>
<div id="response"></div>

function misha_filter_function(){
	$args = array(
		'orderby' => 'date', // we will sort posts by date
		'order'	=> $_POST['date'] // ASC или DESC
	);
 
	// for taxonomies / categories
	if( isset( $_POST['categoryfilter'] ) )
		$args['tax_query'] = array(
			array(
				'taxonomy' => 'category',
				'field' => 'id',
				'terms' => $_POST['categoryfilter']
			)
		);
 
	// create $args['meta_query'] array if one of the following fields is filled
	if( isset( $_POST['price_min'] ) && $_POST['price_min'] || isset( $_POST['price_max'] ) && $_POST['price_max'] || isset( $_POST['featured_image'] ) && $_POST['featured_image'] == 'on' )
		$args['meta_query'] = array( 'relation'=>'AND' ); // AND means that all conditions of meta_query should be true
 
	// if both minimum price and maximum price are specified we will use BETWEEN comparison
	if( isset( $_POST['price_min'] ) && $_POST['price_min'] && isset( $_POST['price_max'] ) && $_POST['price_max'] ) {
		$args['meta_query'][] = array(
			'key' => '_price',
			'value' => array( $_POST['price_min'], $_POST['price_max'] ),
			'type' => 'numeric',
			'compare' => 'between'
		);
	} else {
		// if only min price is set
		if( isset( $_POST['price_min'] ) && $_POST['price_min'] )
			$args['meta_query'][] = array(
				'key' => '_price',
				'value' => $_POST['price_min'],
				'type' => 'numeric',
				'compare' => '>'
			);
 
		// if only max price is set
		if( isset( $_POST['price_max'] ) && $_POST['price_max'] )
			$args['meta_query'][] = array(
				'key' => '_price',
				'value' => $_POST['price_max'],
				'type' => 'numeric',
				'compare' => '<'
			);
	}
 
 
	// if post thumbnail is set
	if( isset( $_POST['featured_image'] ) && $_POST['featured_image'] == 'on' )
		$args['meta_query'][] = array(
			'key' => '_thumbnail_id',
			'compare' => 'EXISTS'
		);
 
	$query = new WP_Query( $args );
 
	if( $query->have_posts() ) :
		while( $query->have_posts() ): $query->the_post();
			echo '<h2>' . $query->post->post_title . '</h2>';
		endwhile;
		wp_reset_postdata();
	else :
		echo 'No posts found';
	endif;
 
	die();
}
 
 
add_action('wp_ajax_myfilter', 'misha_filter_function'); 
add_action('wp_ajax_nopriv_myfilter', 'misha_filter_function');