What is the ruling of Islamic law regarding how to practice religious rituals (prayer and fasting) and convince children to embrace Islam, while the wife and family remain Christian after one has been convinced of Islam?

One must openly declare their faith unless harm is feared, while performing the obligatory duties and avoiding prohibitions to the best of their ability. One should also communicate with Muslims and religious authorities to arrange for the public declaration of their Islam and to seek protection. Minor children follow the religion of the better of their two parents. As for adult children and the wife, they must be invited to Islam. If a Christian wife remains steadfast in her religion, one is not obligated to separate from her.