Does the New Testament disregard the Old Testament?

The very first book of the New Testament begins with:

The book of the generation of Jesus Christ, the son of David, the son of Abraham.

Without the Hebrew scriptures, this verse would make no sense at all.

Most Christians know that Jesus was the son of God, and the adopted son of Joseph, so who is this "David" person?

And in looking up who David was, one will find that David was actually the son of Jesse, so who is this "Abraham" person?

And what does "son of" mean if it doesn't really mean "son of"? One can't properly understand even this first verse of the Greek scriptures without knowledge of Hebrew idioms.

And why were David and Abraham explicitly singled out? What is so special about them?

(The Bible has been described as the story of one man's family, that man being Abraham (or perhaps of his grandson Jacob). David was one of Abraham's descendants, and was the first King in a dynasty that was prophesied to end with the Messiah.)

If one doesn't know and understand the history contained in the Hebrew scriptures, most of what is presented in the Greek scriptures will either not make sense or be misunderstood.

Not only are the two sections of the Bible linked (the first contains prophecies of events in the second, and the second contains references to events in the first), neither one can be fully understood without the other.

The clear answer is NO, since the New Testament relies on the Old Testament to show God's progressive revelation of Himself culminating in Jesus Christ. Christianity teaches that Jesus is God incarnate. Who is this God? It's the God of Israel described in the Old Testament. The same God, who delivered Israel from the slavery of Egypt around 15th to 12th century BC (The Exodus story), delivered the whole world from the slavery of sin and from the consequences of the Fall of Adam (explained in the New Testament) in the 1st century AD.

The New Testament also shows more clearly the character of the God of Israel as being faithful in his unconditional covenant with the people of Israel, and full of mercy and unfailing love. The New Testament is the continuing saga of this love story, with the God of Israel himself coming into the world by being incarnated in Jesus Christ so we can literally see God with our own eyes (John 1:1-18), and more importantly, see this love exhibited in the flesh that was crucified on the cross for us.

If the New Testament were to stand alone without the Old Testament, it wouldn't make much sense on its own. Or worse, it could be used to justify heresies such as Gnosticism, Marcionism, etc. Christians need both the Old and the New Testament books, and they were bound together since the earliest time in a single canon. That's why in all printed Bibles the two sections are labelled with the names "Old" and "New" to signify the relationship: God's previous covenant mediated through Moses with the people of Israel (recorded in the Old Testament) was "replaced"/"added" with a better one through Jesus so that the Gentiles can be included in the New Covenant (as recorded in the New Testament).

NOTE on "replaced/"added": the exact status of the Mosaic covenant for the Jews is not 100% clear but that the Gentiles are included in the New Covenant is certain (cf. Rom 11) as the New Covenant (which was promised in the OT, cf. Jer 31:31-34) is not only for the Jews but for the Gentiles as well. It was also very clear that Israel was planned to be "the light unto the nations" (cf. Isa 42:6, Isa 49:6, Isa 60:3), and therefore by the end of the OT (the book of Daniel was written in the 2nd century BC), the faithful remnant of Israel was waiting for God's "next move" to fulfill the role of Israel vis a vis the rest of the world ("the nations" = "rest of the world"). This "next move" is the New Covenant.

About the OT teachings, the relevance of some OT teachings for the New Covenant people of God is still a matter of debate, although the Ten Commandments and the Great Commandment, both revealed in the OT, are still binding for Christians today. For more details on the debate, I refer you to books such as the 2010 book Five Views on Law and Gospel.