I will use an English example as a sentence. Let's say today is Thursday.

> What day are you going shopping next week?  
>  I'm going shopping on Sunday.

Which is the correct way to answer in Japanese?  
来週は何曜日に買い物に行きますか？

Is it:

> 日曜日に買い物に行きます。

Or:

> 来週の日曜日に買い物に行きます。
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Do I always have to say 「来週の」when talking about a date next week?
In your case, the topic has been already fixated to 来週(は) in the question, so
you don't have to say 来週 again. Everyone understands you are currently talking
about the next week's plan.

If there were no previous context, 日曜日に買い物に行きます normally refers to 次の日曜日 (
_next_ Sunday), which can be different from 来週の日曜日 (Sunday of the next week)
because many think a 日曜日 is at the end of a week. For example, when 次の日曜日
refers to 3 days from now, 来週の日曜日 may refer to 10 days from now.
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Since the question contains 来週の, it is fine just to say 日曜日に買い物に行きます。

* * *

(Not completely on-topic for the question)

As noted in the comment, there will be some subtle thins in 来週の(/先週の）.

First of all, most (tabular) calendars has Sundays as the left most column.
But(?) many Japanese think Monday is the first day of the week (correctly, I
assume).

Consider this week: 10/31(Sun) - 11/6(Sat).

  * On 10/31: technically 11/6 is 来週の土曜日, but I would not use it to denote 11/6. Most probably 次の土曜日 or even (wrongly) 今週の土曜日.
  * On 11/1: similarly I would not use 先週の日曜日 to mean 10/31; it is most probably understood as 10/24. To mean 10/31, probably just 昨日 or 日曜日.
  * During 11/1 - 11/6: 次の日曜日 denotes 11/7 and 来週の日曜日 denotes 11/14. (*)
  * On 11/7: 次の日曜日 and 来週の日曜日 denote 11/14.

Due to the similar ambiguities mentioned in the comment, I guess 来週の日曜日(*) is
less likely to be used as it is later in the week.
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