"take turns doing" is American English. "take turns to do" is British English. Both are correct, but not all Americans (or Brits) are aware of this difference.

The second one sounds correct. You can take turns cooking, take turns watching the children, take turns doing dishes (washing the dishes). That is how we say it in America. You can also say "we are taking turns cleaning the house", etc.

Personally I would use the second one but the first one is also used often. So, they are both fine. There is no difference in meaning.