I've heard other sincere believers say that since Jesus kept the Laws of Moses we should keep them or at least try and somehow that is following Him. Is this what Jesus was talking about in these passages?
Mathew 5:17-20 Jesus said, Do not think that I came to destroy the law and the prophets. No , I have not come to destroy them , but to fulfil them... whoever breaks the least of these commandment , and teaches men so, he will be called the least in the kingdom....
Mathew 5 verse 20 For I(JESUS) say to you, unless your rightousness exceeds the rightousness of the scribes and the pharisees, you cannot enter the kingdom of heaven.
In verse 17, Jesus said we're teach and do the law of Moses to be great in the kingdom and in verse 20 Jesus said unless your rightousness exceeds the rightousness of the Law you'll in no case enter the kingdom of heaven.
Are we to do both, keep the Law of Moses like Jesus, and keep Jesus words in red . Would that be exceeding the scribes and pharisees rightousness?
The new covenant does not require the physical observations or ordinances anymore but as Apostle Paul said:
1 Corinthians 9:19-23 KJV
For though I be free from all men, yet have I made myself servant unto all, that I might gain the more. [20] And unto the Jews I became as a Jew, that I might gain the Jews; to them that are under the law, as under the law, that I might gain them that are under the law; [21] To them that are without law, as without law, (being not without law to God, but under the law to Christ,) that I might gain them that are without law. [22] To the weak became I as weak, that I might gain the weak: I am made all things to all men, that I might by all means save some. [23] And this I do for the gospel's sake, that I might be partaker thereof with you.
Loving our neighbors is the fulfilling the law. To the Jew they became a Jew so that they may win the Jew and not cause them to stumble. They did observe some of the law for the sake of not becoming a stumbling block to them.
You will see in Acts 15 how they discussed the matter and said that they should not trouble the gentiles by requiring them to observe the law but that they abstain from blood and fornication and idols.
In Acts 21, we see Paul observe the law for the sake of the Jews who were believers but are zealous to the law because they heard the news that he was preaching to the Jews among the gentiles not to circumcise their children anymore and forsake the law of Moses.
Acts 21:20-26 KJV
And when they heard it, they glorified the Lord, and said unto him, Thou seest, brother, how many thousands of Jews there are which believe; and they are all zealous of the law: [21] And they are informed of thee, that thou teachest all the Jews which are among the Gentiles to forsake Moses, saying that they ought not to circumcise their children, neither to walk after the customs.
[22] What is it therefore? the multitude must needs come together: for they will hear that thou art come.
[23] Do therefore this that we say to thee: We have four men which have a vow on them; [24] Them take, and purify thyself with them, and be at charges with them, that they may shave their heads: and all may know that those things, whereof they were informed concerning thee, are nothing; but that thou thyself also walkest orderly, and keepest the law.
[25] As touching the Gentiles which believe, we have written and concluded that they observe no such thing, save only that they keep themselves from things offered to idols, and from blood, and from strangled, and from fornication.
[26] Then Paul took the men, and the next day purifying himself with them entered into the temple, to signify the accomplishment of the days of purification, until that an offering should be offered for every one of them.
Jesus likewise taught his disciples to obey what the scribes and Pharisees tell them because they are in Moses' seat and are in authority. But he said not to do according to their works because they say but don't do.
Matthew 23:1-4 KJV
Then spake Jesus to the multitude, and to his disciples,
[2] Saying, The scribes and the Pharisees sit in Moses' seat:
[3] All therefore whatsoever they bid you observe, that observe and do; but do not ye after their works: for they say, and do not.
[4] For they bind heavy burdens and grievous to be borne, and lay them on men's shoulders; but they themselves will not move them with one of their fingers.
This is how we really are to love others. There's no more shoulds. That's why there are divisions among the churches because of the ordinances and many have been made to stumble because of them. This works for individual Christians.
Romans 14:14-17,19-23 KJV
I know, and am persuaded by the Lord Jesus, that there is nothing unclean of itself: but to him that esteemeth any thing to be unclean, to him it is unclean. [15] But if thy brother be grieved with thy meat, now walkest thou not charitably. Destroy not him with thy meat, for whom Christ died. [16] Let not then your good be evil spoken of: [17] For the kingdom of God is not meat and drink; but righteousness, and peace, and joy in the Holy Ghost. [19] Let us therefore follow after the things which make for peace, and things wherewith one may edify another. [20] For meat destroy not the work of God. All things indeed are pure; but it is evil for that man who eateth with offence. [21] It is good neither to eat flesh, nor to drink wine, nor any thing whereby thy brother stumbleth, or is offended, or is made weak. [22] Hast thou faith? have it to thyself before God. Happy is he that condemneth not himself in that thing which he alloweth. [23] And he that doubteth is damned if he eat, because he eateth not of faith: for whatsoever is not of faith is sin.
When you do not cause to stumble anyone, you have fulfilled the law and therefore you exceed the righteousness of the Pharisees.