What is the relationship between the Old Testament and the New Testament?

The New Testament completes and even modifies some parts from the Old Testament. There is no relation between the two except for the fact that the Old Testament was (and still is) the holy book of Jews, and the New Testament was written after Jesus has come and changed the laws.
The Old Testament has much value as it is the word of God and particularly, for Christian believers as it shows that the Law through its ceremonial and religious practices could not bring perfection. Particularly through the many prophecies about the coming messiah, which were clearly fulfilled precisely in Jesus Christ of Nazareth is it valuable, as well as the many things that are written in it 'for our instruction upon whom the end of the ages has come.'

Many New Testament doctrines are set out in the Old Testament and have abiding relevance such as the character of God as Creator (and many other attributes), the origin of sin (hence the need of a savior), the origin of marriage and of language and government. Many of these 'foundational truths' continue, especially as they are explained further under the Christian dispensation.

Religious practice in the Old Testament was more a matter of external observance of outward ceremonies by a priestly class, whereas the New Covenant or Testament which was specifically a matter of the heart, and clearly spoken of in the Old Testament as being more generally applicable to all.
31 Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah: 32 Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD: 33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people. 34 And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD: for I will forgive their iniquity, and I will remember their sin no more.