The book of 1 Corinthians was written to saved people. They happened to live in Corinth, but they were already saved people nonetheless. (See section 17.0)
1 CORINTHIANS 1:1 Paul, called to be an apostle of Jesus Christ through the will of God, and Sosthenes our brother,
1 CORINTHIANS 1:2 Unto the church of God which is at Corinth, to them that are sanctified in Christ Jesus, called to be saints, with all that in every place call upon the name of Jesus Christ our Lord, both theirs and ours:
as was 2 Corinthians:
2 CORINTHIANS 1:1 Paul, an apostle of Jesus Christ by the will of God, and Timothy our brother, unto the church of God which is at Corinth, with all the saints which are in all Achaia:
Neither 1st nor 2nd Corinthians discuss how anybody is to become saved, because they were already saved. Instead, these two epistles address how to remain saved (in addition to some rebuke, instruction on matters of conduct, and so on).
The church at Corinth had some serious problems. They included, and were not necessarily limited to the following:
Such problems with the Corinthian church were so bad, that Paul considered excommunicating many of that congregation, if not all of them. Paul wrote:
2 CORINTHIANS 13:1 This is the third time I am coming to you. In the mouth of two or three witnesses shall every word be established.
2 CORINTHIANS 13:2 I told you before, and foretell you, as if I were present, the second time; and being absent now I write to them which heretofore have sinned, and to all other, that, if I come again, I will not spare:
The reference to "two or three witnesses" (in verse 1) hearkens back to the proceedings for issuing a death-sentence in the Law of Moses (or, as implemented in the New Covenant, excommunication). Moses wrote:
DEUTERONOMY 17:6 At the mouth of two witnesses, or three witnesses, shall he that is worthy of death be put to death; but at the mouth of one witness he shall not be put to death.
DEUTERONOMY 17:7 The hands of the witnesses shall be first upon him to put him to death, and afterward the hands of all the people. So thou shalt put the evil away from among you.
DEUTERONOMY 19:15 ¶ One witness shall not rise up against a man for any iniquity, or for any sin, in any sin that he sinneth: at the mouth of two witnesses, or at the mouth of three witnesses, shall the matter be established.
Moses wrote of "putting evil away from among [them]" by executing the evildoer, but two or three witnesses were required before such a sentence could be carried out. Unlike Moses, however, Christ's kingdom was not of this world (cf. John 18:36), so the execution of such wrongdoers was not done "in the flesh." Even so, and in the same context, such evil was "put away from among them" by excommunication. This (possible) impending excommunication is what Paul warned the Corinthians about in 2 Corinthians 13:1-2.
Some people try to use 1Corinthians 1:14 and/or 1Corinthians 1:17 to show that baptism is not part of being "born again," and that it is not required for the forgiveness of sins, (and therefore, is not required for salvation itself). Here's what Paul actually wrote:
1 CORINTHIANS 1:11 For it hath been declared unto me of you, my brethren, by them which are of the house of Chloe, that there are contentions among you.
1 CORINTHIANS 1:12 Now this I say, that every one of you saith, I am of Paul; and I of Apollos; and I of Cephas; and I of Christ.
1 CORINTHIANS 1:13 Is Christ divided? was Paul crucified for you? or were ye baptized in the name of Paul?
1 CORINTHIANS 1:14 I thank God that I baptized none of you, but Crispus and Gaius;
1 CORINTHIANS 1:15 Lest any should say that I had baptized in mine own name.
1 CORINTHIANS 1:16 And I baptized also the household of Stephanas: besides, I know not whether I baptized any other.
THIS IS A REBUKE AGAINST A DIVIDED, QUARRELSOME, IMMATURE (i.e., carnal) CONGREGATION!! Especially verse 14: "...I'm glad that I am not the one that baptized you..."
Paul continued in the same context, saying:
1 CORINTHIANS 1:17 For Christ sent me not to baptize, but to preach the gospel: not with wisdom of words, lest the cross of Christ should be made of none effect.
...or, as we say today, "you can lead a horse to water, but you cannot make him drink."
Paul's statement is true: After His Resurrection, Christ sent the apostles forth to preach his word. Jesus said:
MARK 16:15 And he said unto them, Go ye into all the world, and preach the gospel to every creature.
...and Paul's statement at 1Corinthians 1:17 is consistent with this. All Paul could do, was preach to people—he could not make them believe, however. Concerning those who would hear this preaching of the gospel, Jesus said in the very next verse:
MARK 16:16 He that believeth and is baptized shall be saved; but he that believeth not shall be damned.
So then, whereas the apostles were sent for the purposes of preaching the gospel, the hearers were required to believe, and to be baptized. Therefore, 1 Corinthians 1:14,17 does not at all suggest that baptism is unnecessary; these verses are part of a rebuke.
Some people say that 1 Corinthians 12:13 means that we are saved by "spirit baptism" alone, and not "water baptism." It says:
1 CORINTHIANS 12:13 For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit.
It should be kept in mind that, as stated previously in section 22.0, the purpose of 1st and 2nd Corinthians was to instruct the Corinthian church on how to conduct themselves. The two letters (or epistles) were instructions on how to remain saved, not how to become saved.
In chapters 12 through 14 of 1 Corinthians, Paul is ultimately speaking about how they are to behave concerning spiritual gifts. (Read these two chapters to see the context in which 1 Corinthians 12:13 appears.) At this point in scripture, Paul is talking about spiritual gifts. Let's look at the preceding few verses:
1 CORINTHIANS 12:4 Now there are diversities of gifts, but the same Spirit.
1 CORINTHIANS 12:5 And there are differences of administrations, but the same Lord.
1 CORINTHIANS 12:6 And there are diversities of operations, but it is the same God which worketh all in all.
1 CORINTHIANS 12:7 But the manifestation of the Spirit is given to every man to profit withal.
1 CORINTHIANS 12:8 For to one is given by the Spirit the word of wisdom; to another the word of knowledge by the same Spirit;
1 CORINTHIANS 12:9 To another faith by the same Spirit; to another the gifts of healing by the same Spirit;
1 CORINTHIANS 12:10 To another the working of miracles; to another prophecy; to another discerning of spirits; to another divers kinds of tongues; to another the interpretation of tongues:
1 CORINTHIANS 12:11 But all these worketh that one and the selfsame Spirit, dividing to every man severally as he will.
1 CORINTHIANS 12:12 For as the body is one, and hath many members, and all the members of that one body, being many, are one body: so also is Christ.
1 CORINTHIANS 12:13 For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit.
1 CORINTHIANS 12:14 For the body is not one member, but many.
(Continue reading the rest of this chapter.) Paul is talking about gifts of the Holy Ghost, here. These various different gifts of the Holy Ghost, are all gifts of the same Holy Ghost!
Previously, in section 3, we've already seen how in the conversation between Christ and Nicodemus, that they were only speaking about one birth:
JOHN 3:3 Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God.
JOHN 3:4 Nicodemus saith unto him, How can a man be born when he is old? can he enter the second time into his mother's womb, and be born?
JOHN 3:5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
Some of us were "born of water" (baptized) in Jerusalem (Acts 2), some (Paul) near Damascus (Acts 9), others in Samaria (Acts 8), others near Gaza (Acts 8), others at Caesarea (Acts 10), others in/around Thyatiria in Macedonia (Acts 16), and still others, in Idaho (I was!), and others, perhaps near your home town.
The members of Christ, the Church, may all have been born of many different waters—but we are all born of the same Spirit, as stated at 1 Corinthians 12:13.