The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.

Optimal indexes for this query:

ALTER TABLE `order_goods` ADD INDEX `order_goods_idx_order_id_goods_id` (`order_id`,`goods_id`);
ALTER TABLE `order_info` ADD INDEX `order_info_idx_order_status` (`order_status`);

The optimized query:

SELECT
        g.goods_id,
        o.order_status,
        SUM(IF(o.shipping_status = 0,
        g.quantity,
        0)) AS 'pending',
        SUM(IF(o.shipping_status = 3,
        g.quantity,
        0)) AS 'picking',
        SUM(IF(o.shipping_status = 5,
        g.quantity,
        0)) AS 'shipping' 
    FROM
        order_info AS o 
    LEFT JOIN
        order_goods AS g 
            ON g.order_id = o.order_id 
    WHERE
        o.order_status IN (
            1, 5, 6
        ) 
    GROUP BY
        g.goods_id 
    ORDER BY
        NULL