First of all, is it convincing that Peter stood up on gave this speech? Whether he gave the speech or not would have been a matter of public record, for this was a crowded international event. People over the known world would have been witnesses to such. So if he hadn't spoken, everyone would have know it.

Yet he says that thousands of Jews believed and got baptized that day. If this didn't happen, everyone would have known it. But it also had to have happened, else the church would have never started. Where did the church come from? And if these masses turned to Christ at such events, they would have known what Peter said at the time. If they knew Acts chapter 2 was lying, they wouldn't have believed. And thus not only can we say that Peter gave such a sermon, but also that the record of his propositions was accurate.

Consider what Peter said. He claimed that Jesus did miracles among them. (Acts 2:22) Now if Jesus hadn't done miracles among them, the crowd would have know that and thus wouldn't have believed. Thus their response testifies that Jesus in fact did miracles among them.

He claims that Jesus rose from the dead, and they didn't dispute that fact either. For the evidence of this was no doubt already known among them. Apparently they too recognized that the story of the disciples stealing the body when the guards were asleep was not believable.

The message they received claimed to be based not simply upon propositional truth, but upon historical facts, which apparently they were incapable of denying. Yet realize that Christianity was not a welcomed message among the religious establishment there, as Christ's crucifixion proved. Yet all the religious leaders had to do was simply to point out that Jesus didn't do miracles and that he didn't raise from the dead. Yet their inability to do so again testifies of the truth of Peter's statements.

It was quite apparent at this point that those in verse 37 who asked "what shall we do?", essentially were saying "I believed, therefore what should I do?" Repentance and baptism shouldn't be viewed as requirements to be saved in addition to faith in Christ. Rather these reveal the quality of one's faith, giving an indication of its salvific value. The faith that saves in an application oriented faith. Peter is saying, "If you truly believe, then prove it by repenting and getting baptized." 

There are those today who view water baptism much as the legalistic Jews viewed circumcision. They view the ritual done in the flesh as having salvific value. Later we see such Jews claiming, "Unless you are circumcised, according to the custom taught by Moses, you cannot be saved." Acts 15:1. These are no different than those in the Christian community who claim "Unless you are water baptized, you cannot be saved". Such is the group of the Baptism who claim you have to get wet to get saved - that which is done in the flesh by hands of men. Rather just as Abraham "received the sign of circumcision, a seal of the righteousness that he had by faith while he was still uncircumcised.' (Romans 4:11), so too those who have a righteousness by faith in Christ, even while still unbaptized, receive the sign of water baptism symbolizing such righteousness. (See also Baptism and the interpretation of Mark 16:16)

But how does "for the remission of sins" fit into this? For if they were already righteous, they would have had their sins forgiven before getting wet. Unless, of course, that this means "in view of the fact that their sins had been forgiven". And in fact this is how A.T Robertson, a well known Greek scholar, interprets it.

Robertson's Word Pictures
{Unto the remission of your sins} (\eis aphesin tôn hamartiôn hûmôn\). This phrase is the subject of endless controversy as men look at it from the standpoint of sacramental or of evangelical theology. In themselves the words can express aim or purpose for that use of \eis\ does exist as in 1Co 2:7 \eis doxan hêmôn\ (for our glory). But then another usage exists which is just as good Greek as the use of \eis\ for aim or purpose. It is seen in Mt 10:41 in three examples \eis onoma prophêtou, dikaiou, mathêtou\ where it cannot be purpose or aim, but rather the basis or ground, on the basis of the name of prophet, righteous man, disciple, because one is, etc. It is seen again in Mt 12:41 about the preaching of Jonah (\eis to kêrugma Iôna\).  They repented because of (or at) the preaching of Jonah. The illustrations of both usages are numerous in the N.T. and the _Koiné_ generally (Robertson, _Grammar_, p. 592). One will decide the use here according as he believes that baptism is essential to the remission of sins or not. My view is decidedly against the idea that Peter, Paul, or any one in the New Testament taught baptism as essential to the remission of sins or the means of securing such remission. So I understand Peter to be urging baptism on each of them who had already turned (repented) and for it to be done in the name of Jesus Christ on the basis of the forgiveness of sins which they had already received. 
 

Furthermore we notice in Acts 10:43 Peter again preaching the gospel says, "that through his name whosoever believeth in him shall receive remission of sins." without making reference to baptism. In fact those in Acts 10 he preached to received the Holy Spirit before they got wet! They received it upon hearing and believing the message. Yet such would not have been possible if we interpret Acts 2:38 to mean that you have to get water baptized before you can receive the Holy Spirit. This again proves that water baptism is not a requirement for salvation, but rather an affirmation  and identification of one's position in Christ.

Notice also this phrase is used elsewhere in the New Testament. It is used of John's baptism. Mark 1:4 "John did baptize in the wilderness, and preach the baptism of repentance for the remission of sins" That was before Christ died.  If their sins were actually remitted in that baptism then there was no need for Christ to die. Furthermore Jesus also used this phrase mentioning nothing about water. Matthew 26:28  "For this is my blood of the new testament, which is shed for many for the remission of sins." So how are all these to be understood?

Mark 1:4 - In light of the remission of sins to come through the blood of Christ
Matthew 26:28 - In light of the remission of sins to follow the blood of Christ.
Acts 10:43 - In light of the remission of sins received through faith in the blood of Christ.

But some replace the blood of Christ with the water of baptism. Some put their faith in the blood of Christ. Others put their faith in the water of baptism.

Romans 3:25 "God hath set forth (Christ) to be a propitiation through faith in his blood, to declare his righteousness for the remission of sins that are past, through the forbearance of God;"

God has made salvation a synergistic process. Yes, having been saved, salvation is monergistically secure. But coming to salvation is synergistic. It involves cooperation. "Save yourselves from this corrupt generation" by putting your faith in Christ as evidenced by your repentance and baptism. Yes, you can be involved in saving yourselves. That's what the Bible teaches. You don't have to give in to fatalistic concepts.