Old Covenant, the Old Testament (OT) law: the 10 Commandments and the law of Moses;

The transition between the Old Covenant and the New Covenant; and the

New Covenant, the New Testament (NT) Commandments: Keep Jesus' "The 8 Commandments".

(Neh 10:29 ... to walk in God's law, which was given by [through] Moses the servant of God, and

to observe and do all the [components of the law listed below]
 
commandments of the LORD our Lord, and
 
[the "10 commandments", written by God on 2 stone tables [tablets] (also see Deu 10:1-5)
 
the "words of the covenant" (Exo 34:28) ]
 
his judgments and his statutes;  (end of Neh 10:29) (also see Mal 4:4)
 
[the "law of Moses", written by Moses in a book (also see Deu 31:9, Deu 31:24-26)]
 
contains all the details for Israel and the Levitical priesthood
 
The old covenant was a covenant between God and the children of Israel (in Horeb (in the desert ), after leaving Egypt)  (see Deu 5:2, Deu 5:6)  (also see Deu 5:31, Deu 6:1, Deu 8:11)

For if that first [old] covenant had been faultless, then should no place have been sought for the second [new covenant].  (Heb 8:7)

The Old Covenant (administered by the Levitical priesthood) was in force until the Resurrection of Jesus.
 
The Levitical priesthood sacrificed many lambs.  Jesus (the Son of God) was the last "lamb" that needed to be sacrificed.
 
By the which [And by that] will [,] we are sanctified [made holy] through the offering [sacrifice] of the body of Jesus Christ once for all.  (Heb 10:10)

Heb 10:11  And every priest [under the old covenant] standeth daily ministering and offering oftentimes the same sacrifices, which can never take away sins:  (Heb 10:11)

Heb 10:12  But this man [Jesus Christ], after he had offered one sacrifice [his own sinless body (see Heb 4:15)] for sins for ever, [he, Jesus Christ, after his resurrection] sat down on the right hand of God [as our high priest under the new covenant];  (Heb 10:12)

 
Jesus the Christ then became our heavenly high priest after the order of Melchisedec.  (Heb 4:14, Heb 5:6 Heb 5:10)   (also see Heb 9:11, Heb 9:24)
 
Note that the order of Melchisedec preceded the Levitical priesthood (Heb 7:1-3), so the order of Melchisedec is not a continuation of (or successor to) the Levitical priesthood.  (Heb 7:11)

Therefore, Christianity is not a continuation of (or successor to) the Levitical priesthood.
 

Christianity has no earthly priesthood, whether molded after the Levitical priesthood or not.  There are no earthly sacrifices remaining for an earthly priesthood to make.  Jesus Christ's crucifixion was the final sacrifice.  (Heb 9:11-12, Heb 9:24)

Christianity has only a heavenly high priest, after the order of Melchisedec: Jesus Christ, the Son of God, resurrected on the 3rd day, a unique event in the history of mankind.  That is why Jesus Christ (the risen Son of God) is the only one that can offer us salvation from our sins.  (Heb 5:9, Heb 9:28, Joh 14:6)

 
Jesus Christ is the mediator of the New Covenant
 
But now hath he [Jesus Christ] obtained a more excellent ministry, by how much also he [Jesus Christ] is the mediator of a better covenant [the new covenant], which was established upon better promises.  (Heb 8:6)

And for this cause [reason] he [Christ] is the mediator of the new testament [new covenant], that by means of death [now that Christ has died as a ransom], for the redemption of the transgressions [sins] that were under the first [old] testament [old covenant], they which are called might receive the promise of eternal inheritance.  (Heb 9:15)

 
With a new high priest and a New Covenant, the Old Covenant then vanished away over the next 40 years, until finally the temple at Jerusalem was destroyed and the Jews scattered (including the Levitical priesthood) in 70 A.D.
 
Then said he [Jesus Christ], Lo, I come to do thy will, O God.  He [God] taketh away [sets aside] the first [old covenant], that he [God] may establish the second [new covenant].  (Heb 10:9)

In that he [God] saith, A new covenant, he [God] hath made the first [covenant] old.  Now that which decayeth and waxeth old is ready to vanish away.  (Heb 8:13)

For the priesthood being changed, there is made of necessity a change also of the law.  (Heb 7:12)
 
For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:  (Heb 8:10)

For I will be merciful to their unrighteousness, and their sins and their iniquities will I remember no more.  (Heb 8:12)

And to Jesus the mediator of the new covenant, and to the blood of sprinkling, that speaketh better things than that of Abel.  (Heb 12:24)

 
Jesus:  If ye keep my commandments [under the new covenant], ye shall abide in my love; even as I have kept my Father's [God's] commandments [under the old covenant], and abide in his love.  (Joh 15:10)
 
Jesus Christ's Commandments for Christians under the New Covenant
 
Note:  For a 1 page summary of this subject, go to
 
Keep "The 8 Commandments" of the New Covenant.
 
Righteousness Through Faith  (we can be Justified by Faith, counted as Righteousness) (see Rom 4 and especially Heb 11 for many examples of faith)
 
But now [under the new covenant] the righteousness of God without the law is manifested, being witnessed by the law and the prophets;  (Rom 3:21)
 
I do not frustrate the grace of God: for if righteousness come by the law, then Christ is dead in vain.  (Gal 2:21)

Is the law then against the promises of God? God forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law.  (Gal 3:21)
 

And be found in him [Christ],
not having mine own righteousness, which is of the law,
 
but that which is through the faith of Christ, the righteousness which is of God by faith:  (Phi 3:9)
 
For the promise, that he should be the heir of the world, was not to Abraham, or to his seed, through the law, but through the righteousness of faith.  (Rom 4:13)

For Christ is the end of the law for righteousness to every one that believeth.  (Rom 10:4)

For Moses describeth the righteousness which is of the law, That the man which doeth those things shall live by them.  (Rom 10:5)

But the righteousness which is of faith speaketh on this wise ... The word is nigh [near] thee, even in thy mouth, and in thy heart: that is, the word of faith, which we preach;  (Rom 10:6-8)

 
That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him [Jesus] from the dead, thou shalt be saved.  (Rom 10:9)

For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation.  (Rom 10:10)

For the scripture saith, Whosoever believeth on him [Jesus] shall not be ashamed.  (Rom 10:11)

For there is no difference between the Jew and the Greek: for the same Lord [Jesus] over all is rich unto all that call upon him [Jesus].  (Rom 10:12)

For whosoever shall call upon the name of the Lord [Jesus] shall be saved.  (Rom 10:13)