How can a girl regain her family's trust after they discovered her involvement in forbidden things, and how can she develop her relationship with Allah after her intention to repent and her commitment to prayer and memorizing the Quran?

Praise be to Allah for your realization of the error in your previous path. Correcting this path begins with sincere repentance to Allah Almighty. If Allah loves you, He will grant you acceptance on earth. Your family members have done well by advising you, for they are responsible for you, as Allah Almighty says: "Allah instructs you concerning your children," and His saying: "O you who have believed, protect yourselves and your families from a Fire." The rectification of your condition and your turning to your Lord is the way to gain your family's trust. Be diligent in observing acts of obedience, especially the obligatory ones, and in avoiding forbidden things, especially major sins. You have done well in establishing prayer and memorizing the Quran, so persevere in that and ask Allah for more.