This Grace thing sounds awesome, but how do I know that I believe and that I have received?

There are several things that are evidence of belief.

Jesus told us that all the commandments hang on love:

Matthew 22:36-40 (ESV)
“Teacher, which is the great commandment in the Law?”
And he said to him, “You shall love the Lord your God with all your heart and with all your soul and with all your mind. This is the great and first commandment. And a second is like it: You shall love your neighbor as yourself. On these two commandments depend all the Law and the Prophets.”

I find it very important that these commandments speak to the state of our heart, which is more important than just our outward actions.

Also, he tells us the types of things that are “the works of the flesh” and that “those who do such things will not inherit the kingdom of God”:

Galatians 5:13-21 ESV
For you were called to freedom, brothers. Only do not use your freedom as an opportunity for the flesh, but through love serve one another. For the whole law is fulfilled in one word: “You shall love your neighbor as yourself.” But if you bite and devour one another, watch out that you are not consumed by one another. But I say, walk by the Spirit, and you will not gratify the desires of the flesh. For the desires of the flesh are against the Spirit, and the desires of the Spirit are against the flesh, for these are opposed to each other, to keep you from doing the things you want to do. But if you are led by the Spirit, you are not under the law. Now the works of the flesh are evident: sexual immorality, impurity, sensuality, idolatry, sorcery, enmity, strife, jealousy, fits of anger, rivalries, dissensions, divisions, envy, drunkenness, orgies, and things like these. I warn you, as I warned you before, that those who do such things will not inherit the kingdom of God.

So we want to be sure that we have believed and received, because Christ came to save because “those that do not believe are already condemned”:

John 3:17-21 ESV
For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. And this is the judgment: the light has come into the world, and people loved the darkness rather than the light because their works were evil. For everyone who does wicked things hates the light and does not come to the light, lest his works should be exposed. But whoever does what is true comes to the light, so that it may be clearly seen that his works have been carried out in God.”

If we believe, if we have Christ in us, we will bear fruit:

Galatians 5:22-25
But the fruit of the Spirit is love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, self-control; against such things there is no law. And those who belong to Christ Jesus have crucified the flesh with its passions and desires. If we live by the Spirit, let us also keep in step with the Spirit.

If we have Christ, we are not to live by the flesh, we are to live by His Spirit, because the Christ in us is greater than the flesh and evil that is in this world:

1 John 4:3
Little children, you are from God and have overcome them, for He who is in you is greater than he who is in the world.

And if we say that we have Christ in us, but are still a slave to the flesh, are we not calling Christ a liar? Isn’t continued rebellion without repentance blaspheming the Holy Spirit that we say lives in us?

Mark 3:29
but whoever blasphemes against the Holy Spirit never has forgiveness, but is guilty of an eternal sin

So let us be sure, and prove ourselves in our belief, let us repent if we say that we have Christ in us.