Will I become free from reincarnation if I forgive everyone who has wronged me and ask forgiveness from everyone I have wronged?

No. According to Hinduism, freedom from reincarnation is possible only on the attainment of Mukti or Moksha. What is Mukti/Moksha? Though different philosophies have different definitions, they agree that Mukti is assured with the vision of God. Forgiving everyone who has wronged you and asking forgiveness from everyone you have wronged is a very good step. But it is not enough because it contains many loopholes like: a) You remember only this life. What about your earlier lives? b) Your definition of 'wrong' itself will start to change, once you start progressing in spiritual life. Then what? and many other loopholes.