What should I do: Should I tell my mother the truth about my father traveling with his second wife, or should I continue to lie based on my father's request, and what is the ruling on my action?

Concealing a secret and not disclosing it is a religious obligation, provided that its concealment does not entail a religiously prohibited matter that infringes upon the rights of the mother, for example. If the questioner is asked about it, she can use allusive language (ma'areedh). If she is unable to do so, there is no sin in lying for the sake of benefit. However, if concealing the secret results in harm to the mother, then there is no sin in disclosing it.