正确说法是let's go swimming next mondaylet's 就有“一起”的意思,因此加上together就重复了而且“下星期一我们一起去游泳吧”这句话是祈使句,不应用疑问语气,也不应用反问语气
Let's go to swim together next Monday.
Shall we go swimming next Monday?
Let's go swimming together next Monday.