What does the story of each patriarch show about our relationship with God?

Abraham teaches us about serving God through strength, perseverance, trust and kindness. Isaac teaches us about serving God through holiness.

Jacob teaches us about serving God in the face of adversity.

Abraham teaches us about serving God through strength, perseverance, trust and kindness. Isaac teaches us about serving God through holiness.

Jacob teaches us about serving God in the face of adversity.

Abraham (18th century BCE) came from ancestry that had been aware of God a couple of centuries earlier but had afterwards slipped into idolatry (Joshua 24:2). Nimrod, the idolatrous tyrant, had brought Abraham's father (Terah) from the Semitic ancestral seat near the conjunction of the Balikh and the Euphrates, and instated him in a position of power in his army in the royal Babylonian city of Ur, where Abraham was born. Nimrod persecuted any who would question his idolatrous cult.
The Kuzari (Rabbi Judah HaLevi, 1075-1141) states that Abraham was gifted with high intelligence; and, as Maimonides (1135-1204) describes, Abraham didn't blindly accept the ubiquitous idolatry. The whole populace had been duped, but the young Abraham contemplated the matter relentlessly, finally arriving at the conclusion that there is One God and that this should be taught to others as well. This is what is meant by his "calling out in the name of the Lord" (Genesis ch.12).

While still in Ur, he remonstrated with passersby in public, demonstrating to them the falsehood of their idols; and our tradition tells how he was threatened and endangered by Nimrod.
Subsequently, Terah relocated to Harran; and it is here that Abraham began to develop a circle of disciples (Rashi commentary, on Genesis 12:5).

Later, God told Abraham in prophecy to move to the Holy Land, which is where Abraham raised his family.
He continued his contemplations, eventually arriving at the attitudes and forms of behavior which God later incorporated into the Torah given to Moses.

Abraham, with God's help, singlehandedly trounced the supremacy of the evil Nimrod.

He received God's promise of inheriting the Holy Land (Genesis 13).

He strove to raise a family (Genesis ch.15, 17, and 24) which would serve God (Genesis 18:19); and God eventually blessed his efforts, granting him many children (ibid., ch.16, 21 and 25), as He had promised (Genesis ch.17).

Abraham founded the Jewish people and lived to see his work live on in the persons of Isaac and Jacob; and he taught many other disciples as well (Talmud, Yoma 28b).

He saved the population of the south of Canaan from invading foreign kings (Genesis 14); and he was feared by neighboring kings (ibid., ch.12 and 20).

Abraham gave tithes (Genesis ch.14), entered into a covenant with God (Genesis ch.15 and 17), welcomed guests into his home (Genesis ch.18) unlike the inhospitable Sodomites (Genesis ch.19), prayed for people (Genesis ch.18), rebuked others when necessary (Genesis ch.20), eulogized and buried the deceased (Genesis ch.23), and fulfilled God's will unquestioningly (Genesis ch.22). He became renowned as a prince of God (Genesis 23:6).

All of these forms of behavior were based upon the ways of God, which Abraham comprehended through his contemplations. These, and similar personality traits, were the teachings of Abraham and his descendants (unlike idolatry, which had no moral character; with worship of the gods accompanied by things such as human sacrifice, "sacred" prostitution, and animal worship).

It is therefore clear why God expresses His love for Abraham (Isaiah 41:8) and calls Himself the God of Abraham (Genesis 26:24), and says that Abraham obeyed Him fully (Genesis 26:5). And this is why Abraham is credited with having begun the religion which became known as Judaism. (However, Abraham and his descendants observed their traditions voluntarily, until the Giving of the Torah to Moses 3325 years ago, when God made it obligatory.)

Isaac is the second of the three Forefathers and (like Abraham and Jacob) one of the greatest personages in Judaism.
Isaac (Yitzhak) was a son of Abraham (Genesis ch.21); and God said that His covenant with Abraham would continue with Isaac (Genesis 17:18-19). Isaac had facial features that were identical to Abraham (Rashi commentary, Genesis 25:19).

God tested Abraham by asking him to offer Isaac as an offering (see Genesis ch.22); and this readiness for self-sacrifice bestowed a special holiness upon Isaac (Talmud, Yevamot 64).
Isaac married Rebecca (Genesis ch.24), his second cousin. Rebecca gave birth to two sons, Esau and Jacob (Genesis ch.25).

God spoke to Isaac, confirming His covenant with him (Genesis ch.26).
Despite harassment from the Philistines, Isaac enjoyed great success (ibid.). He reopened all the wells which his father had dug and the Philistines had stopped up (ibid.). The king of the Philistines, unable to ignore Isaac's phenomenal success, petitioned him to make a mutual treaty of non-aggression (ibid.).

In his advanced years, Isaac sought to bless his elder son Esau (Genesis ch.27). Rebecca took pains to thwart this and to ensure that Jacob would receive the blessing, since she had learned through prophecy (Genesis 25:23) that Jacob would be more favored by God.

Later it became apparent that Isaac admitted to Rebecca's having done the right thing, since he blessed Jacob again (Genesis 28:1-4) even after he found out what had happened (see Genesis 27:6-16).

Isaac sent Jacob to the ancestral family seat in Mesopotamia to choose a wife (Genesis ch.28). Though Jacob didn't return for over twenty years (Genesis 31:38), God granted Isaac extra years, so that he lived on for two decades after Jacob's return.
Isaac was buried next to his father Abraham, in Hebron (Genesis 49:31).

Jacob, son of Isaac, was the last of the Patriarchs (Talmud, Berakhot 16b). He spent a lot of time in the tents (Genesis 25:27) studying his ancestors' teachings (Rashi commentary, ibid.); and eventually, like Abraham and Isaac before him, attained prophecy, in which God confirmed His covenant and promised His protection (Genesis 28:10-15).

When the opportunity presented itself, Jacob asked his elder twin brother Esau to sell him his birthright (Genesis 25:29:34), since he sensed that Esau wasn't pious enough to fully deserve it. Thus began the fulfillment of the prophecy which Rebecca had heard, that Jacob would become the dominant of her two sons (Genesis 25:22-23).

Jacob's life was replete with tribulations (Rashi commentary, Genesis 43:14). He managed to come out ahead despite the wiles of the deceitful Laban (Genesis 29:25 and 31:41) and the danger presented by the angry Esau (Genesis 27:41 and 32:12).

There were painful events with his daughter Dinah (Genesis 34:1-7) and with his being separated for two decades from his beloved son, Joseph (Genesis ch.37); and his wife Rachel died at a young age in childbirth (Genesis 35:16-19).

These troubles were a portent for the tribulations of the Jews in their times of exile. But Jacob received God's affirmation of His covenant and blessing (Genesis 28:13-14; 35:9-12; 46:2-4), signifying that the exile would eventually end. God gave Jacob the honor-title of Israel, indicating his eventual ascendancy (Genesis ch.35).

These three, Abraham, Isaac and Jacob, were the founders of the Israelite (Jewish) people, both physically and spiritually; and God calls himself "the God of Abraham, Isaac and Jacob" (Exodus ch.3) no fewer than eighteen times in the Torah.
It is their ways and traditions that the Torah preserves.

Link: Jewish timeline