In the Name of Allah, Most Gracious, Most Merciful
Praise be to Allah. May the peace and blessings of Allah shower upon our Beloved Messenger, his family, companions, and those who follow them.
I commend you for striving to wear your hijab in the best way possible.
What you must distinguish between are cultural modes of hijab and Islamic standards for hijab. While some cultural modes of hijab are fine if they meet Islamic standards, other cultural ways of covering may need to be adapted to further meet Islamic criteria.
For example, I am familiar with the cultural way of covering in Pakistan. Sometimes this means that a woman will cover but she will wear a thin, transparent scarf that opens to reveal the hair and neck. This is incorrect. All a woman has to do to ensure that her cover accords with Islamic standards is to make sure that her scarf is opaque, meaning not see-through, and thick enough to cover the hair and neck. She should also make sure that her scarf is fastened in such a way that it doesn't come open to expose her neck and chest.
The fact that your family doesn't wish you to cover properly is of no concern. Your obligation is to obey Allah by covering yourself properly when you go out. You cannot elevate your family's wishes above Allah's command. Be kind and respectful to your family, but also be firm about your hijab.
Our understanding of hijab comes from two verses in the Qur'an. The first in Surat al-Nur reads, "And say to the believing women that they should lower their gaze and guard their modesty; that they should not display their beauty and ornaments except what must ordinarily appear thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, or their brothers' sons or their sisters' sons, or their women or the servants whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex, and that they should not strike their feet in order to draw attention to their hidden ornaments. And O you Believers, turn you all together towards Allah, that you may attain Bliss". (Qur'an 24:31).
The other verse in Surat al-Ahzab says, "O Prophet, tell your wives and your daughters and the women of the believers to draw their cloaks close round them (when they go abroad). That will be better, so that they may be recognized and not annoyed. Allah is ever Forgiving, Merciful". (Qur'an 33:59) There are many hadiths about hijab. One of the most famous is the following, "Aisha, (may Allah be pleased with her) reported that Asma, the daughter of Abu Bakr (may Allah be pleased with him) came to the Messenger of Allah (may peace and blessings be upon him) while wearing thin clothing. He approached her and said: 'O Asma! When a girl reaches the menstrual age, it is not proper that anything should remain exposed except this and this. He pointed to the face and hands." [Abu Dawud] Our scholars are unanimous that when a woman reaches the age of puberty, she must cover all her body except her face and hands (and feet for some Hanafis) when she leaves the house or is in the company of non-mahram males. Depending on the culture, hijab has taken many forms. For example, Iranian women wear the chador, Arab women wear the abaya, Pakistani women wear the shalvar qamiz, Malaysian women wear the tudung. There are a lot of ways to cover provided that the woman covers her hair, neck, ears, and chest with a khimar or head veil, and wears some type of long, loose-fitting clothing over the rest of her body. She must not wear anything which is transparent, tight-fitting, or flashy. However, she may wear different colors.
You asked about covering strands of hair. Yes, it is necessary to cover every strand of hair. But depending on how low your hairline is or if you have sideburns, it may be necessary to wear an under-scarf so those little strands won't peek out.