In Moulin Rouge!, Was it the actors recording those songs, or vocal doubles?
Yes, both of them did their own singing for the movie. Ewan McGregor is quoted in this MTV article talking about the pressure that he felt, as it was more singing than he had done previously for parts, and touching on some of the singing training that they did specifically for the movie, and this excerpt from Nicole Kidman pays tribute to how she felt about his singing.
There is a surprising dearth of interviews that I could find where Nicole talks about her singing parts, but I was able to find an interview with Baz Luhrmann talking about how they filmed the singing. They used a traditional method of singing in studio and then playing it back while filming, however they also used a new (at the time) technique of singing it live, and then digitally sampling the recording in later.
However, at least according to semi reliable sources, not all the actors sang their own parts. This article on Buzzfeed says that Jim Broadbent had an opera singer imitating his voice to be dubbed in later.