What does the Bible say about forgiveness and how we learn to forgive effectively?
I gather from the whole question that you are talking about us forgiving others, rather than how to obtain God's forgiveness. There are a number of passages on this subject.
Jesus said that our forgiveness is dependent on our forgiving. "And forgive us our debts, as we forgive our debtors. And lead us not into temptation, but deliver us from evil: For thine is the kingdom, and the power, and the glory, for ever. Amen. For if ye forgive men their trespasses, your heavenly Father will also forgive you: But if ye forgive not men their trespasses, neither will your Father forgive your trespasses." (Matt 6:12-15; also Mk 11:25-26) He repeated essentially the same thing in Luke 6: 37: "Judge not, and ye shall not be judged: condemn not, and ye shall not be condemned: forgive, and ye shall be forgiven." Paul also emphasized that idea: "And be ye kind one to another, tenderhearted, forgiving one another, even as God for Christ's sake hath forgiven you." (Eph. 4:32)
Another reason to forgive is given by Paul in 2 Corinthians 2:6-12. He is talking about one who had to receive punishment for sin. He says don't overpunish, but forgive "lest perhaps such a one should be swallowed up with overmuch sorrow." We should forgive so we don't lay a guilt trip on the other person.
But how do we learn to forgive? How do we do it effectively? Jesus gave an example when asked by Peter about forgiveness. The passage is Matt. 18:21-22: "Then came Peter to him, and said, Lord, how oft shall my brother sin against me, and I forgive him? till seven times? Jesus saith unto him, I say not unto thee, Until seven times: but, Until seventy times seven." He then followed that up with a parable to illustrate that God forgives to the same extent we forgive.
To me the best passage on how to forgive is Col 3:12-17. "Put on therefore, as the elect of God, holy and beloved, bowels of mercies, kindness, humbleness of mind, meekness, longsuffering; Forbearing one another, and forgiving one another, if any man have a quarrel against any: even as Christ forgave you, so also do ye. And above all these things put on charity, which is the bond of perfectness. And let the peace of God rule in your hearts, to the which also ye are called in one body; and be ye thankful. Let the word of Christ dwell in you richly in all wisdom; teaching and admonishing one another in psalms and hymns and spiritual songs, singing with grace in your hearts to the Lord. And whatsoever ye do in word or deed, do all in the name of the Lord Jesus, giving thanks to God and the Father by him." Paul appears to be saying that forgiveness is made easier by doing everything we do in the name of the Lord. If we let the word of Christ dwell in us, through study and practice, we will be able to forgive effectively. And effective forgiveness on our part will lead to effective forgiveness on God's part.