If a person believes Jesus and trusts in the Atonement Jesus secured for their salvation then yes they will be forgiven and have eternity with God..
If your dishonesty with your family is still hurting them or affecting them negatively in some way then yes you should go to them and tell them the truth to stop any lingering harm..
It is hard for me to give advice on the specific issue when i do not know how major or minor the issue was about. So you being the person who knows all the finer details are the one who should make the decision on what your next action,, if any,, should be..
May God's Holy Spirit guide you with the wisdom to act according to His will..