I write from an evangelical point of view using the Bible only as my starting point.
Jesus alone born of a virgin
According to Islam, Jesus is only person to ever be born of a virgin :-
And Mary said "Lord, how can I bear a child when no man has touched me?" (See Quran, "The Imrans", Surah 3:45-48)
Of course, the Bible agrees (Injil of Luke 1:34).
Jesus worked miracles
According to the Quran, Jesus "gave sight to a blind man, healed the leper, and gave life to the dead" (see Quran, "The Table", Surah 5:110-112). Again the Bible agrees, Jesus did hundreds, more likely thousands, of miracles of healing, and other miracles, and even rose some from the dead.
Jesus is the Messiah
According to the Quran, Jesus is the Messiah:-
His name is the Messiah, Jesus the son of Mary
(Quran, "The Imrans", Surah 3:45);
But we need to look at the Old Testament, written hundreds of years before the time of Christ to find out where the term "Messiah" is used and what it means - It means "Anointed". Men were anointed with oil when they were chosen by God for a special purpose, especially those chosen to be High Priest and and those chosen to be the King.
Furthermore the Old Testament says there is someone who is "the Messiah". The use of the term "the Messiah" for a coming deliverer is found in only two passages throughout the whole of the Old Testament!
These passage would be an excellent place to learn who he is and why he would be anointed. What would be the special task of the Messiah when he comes?
Why do the Gentiles rage and the people imagine a vain thing? The kings of the earth set themselves, and the rulers take counsel together, gainst the LORD, and against his Messiah, saying, Let us break their bands asunder, and cast away their cords from us (Psalm 2:1-3).
It means, the human race would like nothing better than to throw off the restraints of God's, and his Messiah's, laws and authority: but it is a vain wish. God cannot be thwarted.
I have set my King upon my holy hill of Zion. I will declare the decree; the LORD has said to me: You are my Son, today I have begotten you. Ask of me and I will give you the heathen for your inheritance, and the uttermost parts of the earth for your possession (Psalm 2:6-8).
According to this passage, written about a thousand years before the coming of Jesus Christ, the Messiah will be from the hill of Zion which is in Jerusalem, meaning he will be Jewish. And the Messiah will be the Son of God himself. And we also learn that this Messiah will be believed on by the Gentiles "to the uttermost parts of the earth".
Now for the second passage:
Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy.
Know therefore and understand, that from the going forth of the commandment to restore and to build Jerusalem unto the Messiah the Prince shall be seven weeks, and threescore and two weeks: the street shall be built again, and the wall, even in troublous times.
And after threescore and two weeks shall Messiah be cut off, but not for himself: and the people of the prince that shall come shall destroy the city and the sanctuary; and the end thereof shall be with a flood, and unto the end of the war desolations are determined.
And he shall confirm the covenant with many for one week: and in the midst of the week he shall cause the sacrifice and the oblation to cease, and for the overspreading of abominations he shall make it desolate, even until the consummation, and that determined shall be poured upon the desolate.
His special task would be that he would be "cut off", that is "he would be judicially put to death", but not for himself, that is "not for his own sins" because he didn't have any, but for the sins of others, all his own people, who would put their trust in him. Through his death he would "make reconciliation for iniquity" meaning he would bring peace between God and sinners who put their trust in this Messiah and his death as a sacrifice for their sins.
And after that he would confirm the Covenant of Grace, that we are saved by repentance and faith in Him alone, "with many", not mostly the Jews but non-Jews, throughout the whole world.
(Bible, Old Testament, Daniel 9:24-27, written about 535 years before Jesus was born. 80 years later, the decree to rebuild Jerusalem was given in 458 BC, and the return of the scribe Ezra from Babylon started on 3rd April 458 BC (Gregorian date) (book of Ezra 7:9) which was exactly "70 weeks", 490 years, before the resurrection 3rd April 33 AD (Gregorian date), to the very day).
You shall call his name "Jesus"
A very striking thought is the most well known amongst Muslims - the Quran says he was called "Jesus". Again the Bible agrees, telling us the Angel Gabriel said to Joseph, Mary's betrothed,
You shall call his name JESUS for he shall save his people from their sins. (Injil of Matthew 1:21)
Unfortunately the Quran never explains what the name Jesus means. The Bible shows the Hebrew origin, and we can see , with a bit of knowledge of Hebrew what his name means. The name "Jesus" means "Saviour": Jesus will be the Saviour of his people, all who put their trust in him, both Jew and non-Jew. The name Jesus is the Greek form of "Yeshua", the which is the Aramaic form of "Yoshua" which is Hebrew.
But "Yoshua" is the shortened name: the full Hebrew name was "Yehoshua" (Numbers 13:16) which means "Jehovah is the Saviour". This was the meaning when Oshea the son of Nun was given the name "Yehoshua" by Moses about 1500 years before Jesus was born.
But it can also mean "Jehovah, the Saviour", and this is its meaning when the Angel gave it to The Anointed, the Messiah. This may be what the Angel meant when he said
They shall call his name Emmanuel which means "God with us" (Injil of Matthew 1:23)
because whenever we call him "Jesus", even though we might not realize it, we are calling him "the LORD (Jehovah God), the Saviour". As the LORD says :-
I, even I, am the LORD, and apart from me there is no Saviour.
(Bible, Old Testament, Isaiah 43:11, written about 700 years before the birth of Jesus Christ)
Jesus is the Word of God
And then, amazingly, the Muslims call Jesus "the Word of God" because the Quran says :-
Allah bids you rejoice in a Word from him. His name is the Messiah, Jesus the son of Mary. (Quran, Surah 3:45).
The Bible agrees :-
And the Word became flesh and dwelt amongst us (Injil of John 1:14).
Again, unfortunately, the Quran does not tell us the full meaning of the title, but the Bible does. Earlier in the same passage we read :-
In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was nothing made that was made. In him was life; and the life was the light of men. And the light shines in the darkness; and the darkness understood it not. (Injil of John 1:1-5)
The Word of God is eternal. It is from eternity to eternity. Therefore Jesus is eternal, from eternity to eternity, too.
Jesus, on the Great Day of Judgment
Finally, according to Islam, Jesus will one day come back from Heaven to earth for "the Day of Doom", (Quran, "Ornaments of Gold", Surah 43:61), that is The Day of Judgement, also known as the Day of Resurrection: "on the Day of Resurrection he shall be a witness against them" (against unbelievers) (Quran, "Women", Surah 4:159). Again the Bible says the same, excepting the Bible says Jesus will be the Judge Himself.
Conclusion - do Muslims really know about Jesus?
These teachings of the Quran mean Muslims need to find out everything they possibly can about the Lord Jesus Christ.
Is Mohammad in the Bible?
Muslim apologists claim Mohammad is named in the Song of Solomon 5:16 :-
His mouth is sweetness itself;
he is altogether lovely.
This is my beloved, this is my friend,
daughters of Jerusalem.
Those interested can look at youtube or the Christian website Answeringmuslims.com "Muhammad in Song of Solomon".
They also claim he is referred to as "another comforter" in Injil of John chapters 14 to 16. e.g. John 14:16 :-
And I will pray the Father, and he shall give you another Comforter, that he may abide with you for ever;
As is repeated so many times this is not referring to Mohammad but it is referring to God the Holy Spirit. His disciples were filled with the Holy Spirit at Pentecost just a few weeks after our Lord Jesus said these things.
Maybe some think that Moses is referring to Mohammad when he speaks of another prophet (Deuteronomy 18:15)
The Lord thy God will raise up unto thee a Prophet from the midst of thee, of thy brethren, like unto me; unto him ye shall hearken;
Moses is referring here to Jesus. Maybe Muslims think Jesus was not a prophet and therefore it could not be referring to him. Jesus was much more than just a prophet only but he was a prophet as well, bringing a message from God as well as being the Son of God: Jesus is sometimes spoken of as having three roles, Prophet, Priest and King; our Prophet to proclaim to us the Gospel, that he himself is the promised Messiah/Christ predicted in the Old Testament by Moses, David and all, and to teach us God's ways; our Priest to make a sacrifice of his own body for our sins; and our King because he rose from the dead to reign over us and all creation, forever.
Salvation is of the Jews
On one occasion Our Lord Jesus said something critically important to a non-Jewish woman:
Ye worship ye know not what: we know what we worship: for salvation is of the Jews. (Injil of John 4:22)
Our Lord said this to a Samaritan woman and this makes it very significant.
Why? Because Samaritans were then, and still are today, part Jewish and part Arab.
A scientific investigation was conducted a few years ago which showed that the samaritans have genetic markers in their DNA which show they have both Israelite and non-Israelite ancestors: and that they are descended from a Levitical clan, the Cohanim (eg see https://www.ncbi.nlm.nih.gov/pubmed/25079122). The Levites were descendants of Abraham through Isaac and Jacob/Israel. (In that the people living in Samaria were not pure Israelite they no longer enjoyed the right to access the Jewish area of the Temple of the LORD at Jerusalem and were not treated like fellow Jews by the Jews.)
This mixture of blood of the Samaritans is precisely what we would have expected from our reading of the Old Testament in the Bible (2 Kings 17, whole chapter).
The same scientific study showed the Arabs and other semitic groups are not so closely related to the Jews.
But Jesus spoke to the Samaritan woman as a non-Jew, even though she had some Jewish blood in her. How much less, then, does salvation come from Americans or the English or Arabs or any other Semitic blood group, or any other nation, seeing as this woman, who was part-Jewish by blood, needed to hear about salvation from the Jews!
Even today we still need to hear about salvation from the Jewish Old Testament and the New Testament, mostly written by Jews and those parts we think were not written by Jews (Luke and Acts) were still approved by the Apostles who were Jews. [This can be seen by comparing 1 Timothy 5 v 18, which was written by the (Jewish) Apostle Paul, with Luke 10 v 7, which was written by Luke who was the only Gentile author in the New Testament. The Apostle Paul confirms that the Gospel of Luke is Holy Scripture, the word of God. (The Greek of 1 Timothy 5:18 is closer to the Greek of Luke 10:7 than that of Matthew 10:10.)
But now salvation, through our Lord Jesus, is for all mankind
But the special dealings God had with the Jewish nation were not for their benefit alone, but for the long-term benefit of all nations, all tribes and all language groups, including the Arabs and all the other semitic nationalities.
When God was making a special covenant with Abraham and all his descendants through Isaac and Jacob, Abraham prayed to God
And Abraham said unto God, O that Ishmael might live before thee! (Genesis 17:18)
And God answered Abraham:
And as for Ishmael, I have heard thee: Behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly; twelve princes shall he beget, and I will make him a great nation. (Bible, Old Testament, Genesis 17:20) - Abraham lived about 2000 years before Christ.
What God is promising is that one day some time in the future, maybe the distant future, the descendants of Ishmael would be greatly blessed. But it cannot be that this blessing would be without the only blessing that really matters: salvation and peace with God. How could Abraham be happy if Ishmael's descendants were merely materially blessed but not spiritually?? God was promising to Abraham that one day salvation would come to the descendants of Ishmael.
This salvation is through the death and resurrection of our Lord Jesus, to whom God the Father promised through Isaiah the prophet:
It is too light a thing that you should be my servant
to raise up the tribes of Jacob
and to bring back the preserved of Israel;
I will make you as a light for the nations,
that my salvation may reach to the end of the earth.
(Holy Bible, Isaiah 49:6, written about 700 years before the birth of Christ).
It is not just agreeing with God's holy ways that pleases God: God must punish all sin because he is holy. He must punish all hatred, all cruelty, all selfishness, all self-seeking, all sexual immorality, all lying, all covetousness, all idolatry, and all other sins for
for the LORD (God) sees not as man sees; for man looks on the outward appearance, but the LORD looks on the heart. (Holy Bible, Old Testament, 1 Samuel 16:7)
There can be no forgiveness of sins by God unless God's justice can be satisfied, and his laws fully obeyed. No one can even approach God to pray to him until the problem of our very offensive sinfulness is taken away: God will not receive us to hear our prayers!
Jesus Christ, God the Son, came to earth to obey God's Law, the Law given by Moses, on our behalf, and to take our punishment upon himself on the cross, so we could have peace with God through believing on him.
But he was pierced for our transgressions,
he was crushed for our iniquities;
the punishment that brought us peace was on him,
and by his wounds we are healed.
We all, like sheep, have gone astray,
each of us has turned to our own way;
and the Lord has laid on him
the iniquity of us all.
After this he rose from the dead because it was not possible for death to hold him :-
Therefore I will give him a portion among the great,
and he will divide the spoils with the strong,
because he poured out his life unto death,
and was numbered with the transgressors.
For he bore the sin of many,
and made intercession for the transgressors.
(Bible, Old Testament, (Isaiah 53:5,6,11) - written about 700 years before the birth of Jesus the Messiah.