Do these sentences have the same meaning?
"I go shopping at the weekends."
"I go shopping at weekends."
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Sign up to join this community"At the weekends" implies a little more that you're going shopping every weekend. It's not as strong as directly saying, "I go shopping every weekend", but it has this underlying implication. You might say, you're going shopping not on some weekends, but on the weekends as a category.
In contrast, "at weekends" implies more that if you go shopping, you do it on the weekend and not on any other day of the week. You don't necessarily go shopping every weekend, but if you go, it's probably on the weekend.
This is no clear, strong distinction that's obeyed every time, but more of an implied tendency.
Both are correct, certainly is colloquial use, and mean the same thing.
The Cambridge Dictionary lists "at weekends" as UK and "on (the) weekends" as US.
See also this related question: https://english.stackexchange.com/questions/75466/at-on-the-weekends