So Jesus said that people are saved by believing in him. But just in case you're still not convinced, the Bible is very clear that people are not saved by doing good deeds or obeying God's law.
Here's more scripture:
"Well then, are we Jews in any better condition than the Gentiles? Not at all! I have already shown that Jews and Gentiles alike are all under the power of sin. As the Scriptures say:
'There is no one who is righteous, no one who is wise or who worships God. All have turned away from God; they have all gone wrong; no one does what is right, not even one. Their words are full of deadly deceit; wicked lies roll off their tongues, and dangerous threats, like snake's poison, from their lips; their speech is filled with bitter curses. They are quick to hurt and kill; they leave ruin and destruction wherever they go. They have not known the path of peace, nor have they learned reverence for God.'
Now we know that everything in the Law applies to those who live under the Law, in order to stop all human excuses and bring the whole world under God's judgment. For no one is put right in God's sight by doing what the Law requires; what the Law does is to make us know that we have sinned.
But now God's way of putting people right with himself has been revealed. It has nothing to do with law, even though the Law of Moses and the prophets gave their witness to it. God puts people right through their faith in Jesus Christ. God does this to all who believe in Christ, because there is no difference at all: everyone has sinned and is far away from God's saving presence. But by the free gift of God's grace all are put right with him through Christ Jesus, who sets them free. God offered him, so that by his blood he should become the means by which people's sins are forgiven through their faith in him. God did this in order to demonstrate that he is righteous. In the past he was patient and overlooked people's sins; but in the present time he deals with their sins, in order to demonstrate his righteousness. In this way God shows that he himself is righteous and that he puts right everyone who believes in Jesus.
What, then, can we boast about? Nothing! And what is the reason for this? Is it that we obey the Law? No, but that we believe. For we conclude that a person is put right with God only through faith, and not by doing what the Law commands. Or is God the God of the Jews only? Is he not the God of the Gentiles also? Of course he is. God is one, and he will put the Jews right with himself on the basis of their faith, and will put the Gentiles right through their faith. Does this mean that by this faith we do away with the Law? No, not at all; instead, we uphold the Law."
-Romans 3:9-31
Jesus also told this parable to people who were sure of their own goodness and despised everybody else. "Once there were two men who went up to the Temple to pray: one was a Pharisee, the other a tax collector. The Pharisee stood apart by himself and prayed, 'I thank you, God, that I am not greedy, dishonest, or an adulterer, like everybody else. I thank you that I am not like that tax collector over there. I fast two days a week, and I give you one tenth of all my income.' But the tax collector stood at a distance and would not even raise his face to heaven, but beat on his breast and said, 'God, have pity on me, a sinner!' I tell you," said Jesus, "the tax collector, and not the Pharisee, was in the right with God when he went home. For those who make themselves great will be humbled, and those who humble themselves will be made great."
Some people brought their babies to Jesus for him to place his hands on them. The disciples saw them and scolded them for doing so, but Jesus called the children to him and said, "Let the children come to me and do not stop them, because the Kingdom of God belongs to such as these. Remember this! Whoever does not receive the Kingdom of God like a child will never enter it."
-Luke 18:9-17
So Jesus is very clear that people are not saved by doing good works. We are saved only by believing in him.
This is great news since we are not able to live perfect lives. God demands perfection, and we all fall short of that. If we're not saved by his kindness and grace, then we could never be saved.
The above passage, in the book of Romans, describes salvation as a gift. It's not something that we earn. A gift is accepted, not earned.
Here's more scripture that talks about what Jesus does for those who choose to follow him.
"In the past you were spiritually dead because of your disobedience and sins. At that time you followed the world's evil way; you obeyed the ruler of the spiritual powers in space, the spirit who now controls the people who disobey God. Actually all of us were like them and lived according to our natural desires, doing whatever suited the wishes of our own bodies and minds. In our natural condition we, like everyone else, were destined to suffer God's anger.
But God's mercy is so abundant, and his love for us is so great, that while we were spiritually dead in our disobedience he brought us to life with Christ. It is by God's grace that you have been saved. In our union with Christ Jesus he raised us up with him to rule with him in the heavenly world. He did this to demonstrate for all time to come the extraordinary greatness of his grace in the love he showed us in Christ Jesus. For it is by God's grace that you have been saved through faith. It is not the result of your own efforts, but God's gift, so that no one can boast about it. God has made us what we are, and in our union with Christ Jesus he has created us for a life of good deeds, which he has already prepared for us to do."
-Ephesians 2:1-10