When witnessing to Jewish people, I’ll go over the moral Law and speak of the holiness, righteousness and justice (and judgment) of God as well as the mercy and salvation of God, all from Old Testament passages. I’ll share some Messianic passages in the Old Testament, especially from Psalms and Isaiah. I just share the passages and ask the person to consider Who the passage is really talking about. I’ll go ahead and tell them that Yeshua is the One who has fulfilled all these prophecies. Aside from proclaiming God’s Word, I think that love, compassion and prayer for God’s mercy is most important.
Here is some instruction from John Piper on witnessing to Jewish people.
You could also read parts of their scriptures that talk about the need for forgiveness and the fact that a Messiah is coming who will bear the sins of his people, and see if they would be open to that interpretation. Then talk about the gospel in relationship to those prophesies.
Stress over and over again that when we meet God he is going to be absolutely righteous and absolutely just, and our good deeds will never suffice to save us from judgment: “If you, O Lord, should mark iniquities, O Lord, who could stand? But there is forgiveness with you, that you may be feared” (Psalm 130:3-4).
Then ask them, “What is that forgiveness based on? Why would there have been a whole series of animal sacrifices if no sacrifice were necessary for the forgiveness of sins. And if sacrifice is necessary for the forgiveness of sins, whose sacrifice can cover all of my sins?”
Answer: “Christ bore our sins in his body on the cross that we might die to sin and live to righteousness” (1 Peter 2:24).
So just tell the story over and over again, as often as they’ll listen.
Should you stop sharing the Scriptures if it offends them?
Absolutely not. If they say, “Be quiet and don’t talk to me anymore,” then you should probably stop talking to them. But if they are just offended then you should keep on seeking to speak with them in a way that would be least offensive. You would not do that by changing the content of your message but by having a demeanor that communicates as much love as possible.
Some people think that we don’t need to evangelize Jews because they will still be saved by the old covenant. Is that true?
No. The notion that we don’t need to evangelize Jews because there remain two distinct covenants—a Jewish covenant and a Christian covenant—is simply unbiblical. The New Testament pictures the old covenant as coming to its climax and consummation in Jesus Christ.