custom taxonomies

Try the following, replacing "post_tag" with your custom taxonomy.

add_filter("gform_post_data", "custom_taxonomy", 10, 2);
function custom_taxonomy($post_data, $form){
	//-----------------------------------------
	//Replace "post_tag" with your taxonomy name
	$custom_taxonomy = "post_tag";
	//----------------------------------------

	$tags = implode(",", $post_data["tags_input"]);
	$post_data["tax_input"] = array($custom_taxonomy => $tags);
	$post_data["tags_input"] = null;
	return $post_data;
}

You are right. It doesn't like the multiple tag fields. We will have to get a little creative to get around the problem. The first hook prevents the tags from being inserted into the post tags. The second hook inserts them into the custom taxonomies using the wp function you found. Let me know how it goes

add_filter("gform_post_data", "empty_tags", 10, 2);
function empty_tags($post_data, $form){
    //replace 4 with your actual form id
    if($form["id"] !=4)
        return;

    $post_data["tags_input"] = null;
    return $post_data;
}

add_filter("gform_post_submission", "insert_custom_taxonomies", 10, 2);
function insert_custom_taxonomies($entry, $form){

    //REPLACE 4 with your actual form id
    if($form["id"] !=4)
        return;

    //getting post
    $post_id = $entry["post_id"];

    //REPLACE 2 with your actual field id and "post_tag" with your custom taxonomy
    wp_set_object_terms($post_id, explode(",", $entry[2]), "post_tag");

    //REPLACE 3 with your actual field id and "activities" with your second taxonomy
    wp_set_object_terms($post_id, explode(",", $entry[3]), "activities");
}

Good call on not using the tag field. It does save you that function call. Better documentation for hooks and objects are coming soon.