Luna is Usagi’s cat, or at least in front of others she acts like that, the truth is that she is advisor to her, as she was to Queen Serenity (princess Serenity’s mother) during the time of the Moon Kingdom.
Their relationship begins in the first act of the first arc when Usagi saves Luna from a bunch of bully kids. There she begin following her and then give her the first brooch to transform in to Sailor Moon and not leaving her anymore.
Their relationship at first seems as one of a teacher and unwilling student, because lot of times Usagi just tries to ignore Luna’s advice.
The first time we actually see how much Luna cares about Usagi is when she feels guilty for not noticing/remembering before that it was Usagi who was the princess and not recognizing Endymion in Mamoru, leading to a sad Usagi and a kidnapped Mamoru.
The relationship between the two of them is strong and caring, even if they sometimes seem to argue, when the crisis is in the full explosion, those two will work well together to save the day.
Luna’s partner is Artemis with whom she has a daughter in 30th century