How should I act Islamically towards my mother who had illicit relationships in the past, knowing that I still suffer psychologically from that and it causes me a feeling of shame in front of the neighbors?

If your mother has repented from committing haram acts, then you must move past what has transpired, for repentance erases what came before it. If you find yourself unable to love her because of her past actions, there is no blame upon you, provided that this does not lead you to disrespect her. For love and hatred are matters of the heart that a person does not fully control. Your duty is to honor your mother and treat her kindly, no matter her condition, for the mother's right is immense.