JOEL - The scriptures do mention certain times or things we might do that can bring God's forgiveness.
Our sins are forgiven when we are healed through a priesthood blessing:

"And the prayer of faith shall save the sick, and the Lord shall raise him up; and if he have committed sins, they shall be forgiven him. (James 5:15)

When we pray. After Enos prayed all night:

"there came a voice unto me, saying: Enos, thy sins are forgiven thee, and thou shalt be blessed." (Enos 1: 5)

When we do missionary work:

"Therefore, thrust in your sickle with all your soul, and your sins are forgiven you, and you shall be laden with sheaves upon your back, for the laborer is worthy of his hire. Wherefore, your family shall live. (D&C 31: 5)

When we share our testimony with others:

"Nevertheless, ye are blessed, for the testimony which ye have borne is recorded in heaven for the angels to look upon; and they rejoice over you, and your sins are forgiven you. (D&C 62: 3)

When we confess our sins:

"Nevertheless, he has sinned; but verily I say unto you, I, the Lord, forgive sins unto those who confess their sins before me and ask forgiveness, who have not sinned unto death. (D&C 64: 7)

When we are obedient to God:

"VERILY thus saith the Lord unto you, my servant Lyman: Your sins are forgiven you, because you have obeyed my voice in coming up hither this morning to receive counsel of him whom I have appointed. (D&C 108: 1)

But normally there isn't one single event where we know are sins are forgiven. We need to go through a series of steps that bring us to a realization through the spirit that God has forgiven us. We first recognize that we have sinned; we feel godly sorrow for the sin; we make right what we did wrong; we promise God that we will not do it again, and ask Him for forgiveness. After all this and some time passes we realize that we have overcome the sin and feel a peace in our souls as the spirit witnesses to us that we have been forgiven.