"We want to go shopping."
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."
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.