Help writing a query that select the min column in a group by
You can try this:

SELECT
    u.id,
    u.nickname,
    t.minPrice
    FROM
      user u
      LEFT JOIN user_service us ON us.user_id = u.id
      INNER JOIN (
    SELECT user_id, MIN(price) minPrice
        FROM user_service
        GROUP BY user_id    
    ) t ON t.user_id = u.id AND us.price = t.minPrice
    WHERE us.service_type IN ('TYPE1','TYPE2') and u.id=1
   group by u.id;