Characters will tell the truth as they see it. There should be good reasons for them to believe the way they do (even/especially when this puts them at odds with the rest of their world and other characters).
Lies can be revealed gradually as you read more of a character and realise, over the space of the story, that what they are saying/doing is at odds with what is truthful for that setting. What made that character lie? What are they hoping to gain from it? If it is a question the character really doesn't think they are lying, how do they react when it is proven that they are?
Good story ideas there with lots of potential for conflict.