Both are correct. The differences in the preposition is just American and British English. In British English, the preposition is 'at', and in American English, it's 'on'. In your sentences, the most important difference is the singular and plural form of the word 'weekend'. I'll give examples with the British version. We didn't do much at the weekend / We are going to Paris at the weekend - ONE specific weekend, either the previous one or the next one. We don't usually do anything exciting at weekendS - weekends in general, or all weekends.

We don't really use "on the weekend" much. Instead we say "over the weekend" since the weekend spans more than one day. "On" is used more often to refer to a specific day. For example, "I'm going to brunch on Sunday" or "I'm doing my homework over the weekend." "At the weekend" is incorrect, I think. "At" is used more for physical places or time. For example, "I'm going to my friend's house at one o'clock" or "I'm currently at a party."

In American English we tend to say "on the weekend" or (less commonly) "on (the) weekends." In British English "at the weekend" and "at weekends" are more common.