Specifying a Category From a Post ID

I’m trying to figure out a way to be able to display a category’s name or slug by just using a post ID.

Currently a plugin I’m using does a search for certain posts that contain event meta values. I’d like to be able to display the related category information (name, slug, etc.) for each event

$where_category_clause = (0 == $category) ? '' :
'AND cats.term_taxonomy_id =
'.$wpdb->escape(stripslashes($category));

	SELECT DISTINCT
		meta.meta_value as 'date',
		post.post_title as 'title',
		post.post_content as 'fulltext',
		post.ID as id,
		post.post_excerpt as 'excerpt'
	FROM
		{$wpdb->postmeta} as meta,
		{$wpdb->posts} as post,
		{$wpdb->term_relationships} as cats
	WHERE
		meta.post_id = post.ID
	AND
		post.ID = cats.object_id
	AND
		meta.meta_key = 'rs_event'
	AND
		post.post_date <= '".current_time('mysql')."'
	AND
		meta.meta_value >= {$lower_time}
	AND
		meta.meta_value <= {$upper_time}
	AND
		post.post_status = 'publish'
	{$where_category_clause}
	ORDER BY
		meta.meta_value {$sort_order}

I’m assuming I need to make a seperate query after I’ve retrieved the event (post) ID, but I’m not sure how to go about calling each event’s category. I’m sure it involves JOIN-ing tables, but I’m really unfamiliar with how that works.