It largely depends on which Calvinist you ask :)
The author of one of the more famous Systematic Theology works (can't remember if it was Shedd, Berkhof, or Hodge... I remember reading it in a defense of Classical Arminianism by Picirilli) said that predestination is not unto faith. This indicates that predestination is unto regeneration.
So what is required is a definition of Calvinistic regeneration. Here is Berkhof's take on regeneration (too long to post, please read. Also read the discussion. It's highly interesting.). Essentially, regeneration is a change in a man's condition, from being spiritually dead to being spiritually alive. Once a man is spiritually alive, he has the ability to see his rebellious condition. Because of this, he has an irresistible urge to be saved and have faith in Christ, and he does. Regeneration and salvation are not something that are considered to happen in a chronological order whereby one happens then the other. That regeneration precedes faith and therefore salvation is only a logical ordering. In actuality, the chronological ordering is simultaneous. An important note, though. In Romans, we see that faith comes by hearing, and hearing by preaching. We also see in Ephesians that faith is a gift of God. So, simply put, the faith is given to the believer by God, and the believer comes upon this faith by hearing the Gospel.
There is a second question, how sin affects a regenerate's life. The first point is that regeneration occurs at a point in time. An elect man is born into original sin and rebellion toward God, and regenerated at a later point. He does not necessarily emerge from the womb in a regenerate state.
However, even if this was the case, a regenerated man will still sin! This does not mean that this is accepted by God, but rather that through the man's faith in Christ, he has access to the forgiveness brought through Christ.
It has been asked, "Is faith a gift of God? Does the verse from Ephesians necessitate this interpretation?"
First, whether or not an individual believes faith to be a gift from God is not the question at hand. We're talking about a Reformed interpretation of the doctrine, so for point of this answer, faith is a gift from God.
More importantly though, is it good to consider faith as a gift from God?
A cursory glance at the verse in Ephesians says no.
For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast.
(Ephesians 2:8-9 ESV)
I'm not a Bible languages scholar, but the commentaries I've read all say that "It's" antecedent is "grace," not "faith" because the gender of the words in question. So, we can make the argument from language that "it" refers to "grace," not "faith." However, we have not established that faith is not a gift in this verse, only that the antecedent of the gift is "grace." If we're going to use this information to determine that faith is not a gift, we're going to need to say that grace is a gift from God, but the channel through which grace is applied is not a gift from God and is instead resident within the believer. This is shaky ground for Arminians and Calvinists alike (and Lutherans, etc.).
The Calvinist position is that the mechanism whereby God applies His grace is also a gift, or at least the ability to act on that mechanism. If the believer had present within himself the channel through which Grace is applied, in addition to the ability to employ the channel, and chose to employ this channel when another did not, this would give the believer a right to boast over the non-believer that "I was wise enough to act upon something within me, whereas you were not." The Reformed interpretation of this verse completely blows this argument out of the water: none of salvation is by works, it is all of grace. A common answer to this is that "faith is not a work," but if it is something that the believer does and something an unbeliever does not, then why should we not consider one's faith a work? Without considering faith as a gift of God, we are relegated to consider faith a work and are no better off than demons
In other places too, we see faith as being something God can administer. In Romans, we see that God assigns a measure of faith.
For by the grace given to me I say to everyone among you not to think of himself more highly than he ought to think, but to think with sober judgment, each according to the measure of faith that God has assigned.
(Romans 12:3 ESV)
In Mark, we see that Christ does not rebuke the man who begs his Lord to "help with his unbelief," and rather the Lord blesses the man through the healing in the passage.
And Jesus said to him, “‘If you can’! All things are possible for one who believes.” Immediately the father of the child cried out and said, “I believe; help my unbelief!” And when Jesus saw that a crowd came running together, he rebuked the unclean spirit, saying to it, “You mute and deaf spirit, I command you, come out of him and never enter him again.” And after crying out and convulsing him terribly, it came out, and the boy was like a corpse, so that most of them said, “He is dead.” But Jesus took him by the hand and lifted him up, and he arose. And when he had entered the house, his disciples asked him privately, “Why could we not cast it out?” And he said to them, “This kind cannot be driven out by anything but prayer.”
(Mark 9:23-29 ESV)