What does each patriarch in the Old Testament show about our relationship with God?

Answer 1

The answer lies in the title Patriarch and not in any story. Putting it simply a patriarch is "The male head of family or tribe". As God is the patriarch of the human race, so a father is the patriarch of his family. The wife would be referred to as a Matriarch.

And

I think you can see the same / similar with each story of the patriarchs, so I will just mention Noah.

Like everybody in the world at that time ( and at any time ) Noah did not deserve to be saved by God. We are the same we do not deserve salvation. By God's grace salvation is offered to us so our relationship with God is by God's grace not based on anything we have done. Eph 2:9

In faith Noah believed God and was saved. Believers too have a relationship based on faith. Faith that God has provided a suitable sacrifice for our sin. Rom 3:22,23.

By faith Noah obeyed God and built the ark. Our on going relationship with God is also by faith. As God instructs we too act in faith.

The blessings of God come through obedience. Noah obeyed God and he was blessed. The same with the other Patriarchs, and when they didn't obey God disciplined them - Moses Numbers 20:8-12. It is the same with Christians today blessing comes through obedience.

God is our guide. God guided the patriarchs, God guides believers today.

Our relationship as believers is secure. In the stories of the patriarchs they didn't always do what was right but God worked out His plan through them. We too don't always get it right but because our relationship is not based on what we do God does not abandon us.

Answer 2

In Judaism, Abraham was exemplary in pioneering in righteousness, in originality, patience and fearlessness, and in lovingkindness. Isaac was exemplary in self-sacrifice and purity.

Jacob was exemplary in serving God despite exile and tribulations, and in scholarliness.

See also:

More about the three Patriarchs