"We want to go shopping."
要 will or "going to happen" 想 want or "have a desire but might or might not happen"
想 and 想要 are same in this case "want to do something". EX.我们想买东西。or 我们想要买东西。 But if you are describing "want something". 想要 and 要 are the same. EX.我想要钱。 or 我要錢。I want money. If you are describing"think of " or "think about". 想 is correct. EX.我想你 I miss you. If you are describing "going to". 要 is correct. EX.我们要去买东西。 Moreover, 我们"要"买东西。is correct grammar but ambiguous. It can also mean "We want to go shopping." and "We are going to shopping."
In this particular sentence, there is no difference.
But we should note that 要 has a lot of other meanings, and should not be generalized as "want" in other usages.
I don't know for certain, but it seems like 想 is more appropriate in situations where you want to do something, but you still may not do it. So in this case, the people want to go shopping, but maybe they haven't made up their mind yet. That means 要 would be more appropriate if the decision is more definite, like you're going to do something like 我要订位. And from my experience, it seems like people use 想 more often as they talk.
I saw a bloke on youtube explaining that 想 is sort of theoretical wish, while 要 suggests actual intention to do something. I probably oversimplified it terribly and much depends on the context, but it helped me to start seeing the difference
I'm not an expert, but 要 is often times used to express things that will happen. Imagine 想 here being used in the "we want to go shopping" sense, while using 要 produces "we will go shopping".
As far as I know, 想 is less harsh. Kind of like 'I would like to go shopping'.
From my limited understanding, 要 is to want while 想 is to want to (among more meanings).
Translation should be 我们想去买东西, as 我们想买东西 means we want shopping. in addition, 买东西 sounds like 'buy something', otherwise '购物'(gòuwù) means hopping
The literal translations "We would like to buy something." and "We want to buy something." should be accepted here.