In 1 Corinthians 3, Paul says that he still can not address the Corinthians as spiritual people. Instead, he says they are fleshly, mere infants in Christ. By saying that the Corinthians are fleshly and not spiritual, Paul is not saying that the Corinthians aren't Christians. After all, he describes them as being in Christ and in the previous chapter he said that they have the mind of Christ. Paul acknowledges the Corinthians as the body of Christ. He knows that they received the Spirit. In saying that he can not speak to them as spiritual, neither is Paul saying that the Corinthians aren't religious enough. He is not saying that the Corinthians need to focus more on their "spirituality" as opposed to their every day physical needs. Instead, Paul is driving home the same point that he has been making throughout the letter up to this point: the Corinthians have the Spirit of God among them but they don't yet fully understand what that means for their behavior.
As we've seen in the previous passages in 1 Corinthians, the problem in Corinth is not a lack of zeal for the Spirit or spiritual things. Paul almost certainly uses the term spiritual here to describe what the Corinthians are not precisely because that is how they saw themselves; as deeply spiritual. So when Paul says that the Corinthians are not spiritual but fleshly, it is not so much a comment about the quantity of the Corinthians spirituality as its quality. Paul is not saying that they are unconcerned with "spiritual" things but that they are approaching the Spirit in the wrong way. They are seeking to use the Spirit for their own advancement just like they would use any other tool or philosophy in life. Paul is telling them that even though the Spirit of God is among them they are still acting like the world acts by striving to exalt themselves. Their spirituality is fleshly, that is, immature.
Paul says that the evidence of this fleshly, immature spirituality are the divisions and strife that exist among the Corinthians. If the Corinthians had a mature understanding of the wisdom of God revealed in the cross of Christ, they would realize that a genuine spirituality would mean submitting to each other in unity and common mission. Instead, they are allowing themselves to be fragmented by attaching themselves to different leaders. In all likelihood, the Corinthians were doing this because they saw those leaders as more "spiritual" and more "wise" than Paul so they thought they could advance further in the faith. This in turn causes Paul to address what it means to be a Christian leader.
For Paul, to be a Christian leader is actually to be a servant. He says in v.5 "What then is Apollos? What is Paul? Servants through whom you believed". Whatever authority Paul or Apollos have, they are ultimately still nothing more than servants of God. As servants, they are each given different roles within God's mission. Paul says that he planted the church and Apollos watered it but it always ultimately God who causes the growth. Only God can grow and mature his people. That is not a human task. Paul says "neither the one who plants or the one who waters is anything but only God who gives the growth." Therefore, Christian leadership is not a matter of exalting oneself. It is not a matter of gathering as many followers as you can. It is not even a matter of growing your church since that is something only God can do. The task of Christian leadership is nothing other than submitting oneself to God as a servant, ready to do whatever work God calls us to in his field.
Of course, this is really more than Paul's view of leadership. It flows from his view of the Christian life as an imitation of the pattern that he sees in Jesus; that paradigm which is summed up so well in Philippians 2:5-11. Christ, who is God and had every reason to exalt himself, instead chose to humble himself taking the lowest of low positions, even death on a cross. "Therefore, God exalted him to the highest place and gave him the name that is above every name that at the name of Jesus, every knee should bow in heaven and on earth and under the earth to the glory of God the Father." Paul believes this is what all Christians are called to; this humbling of oneself. This is very thing Paul has been referring to as the wisdom of God.
Of course, this is such a difficult pattern for us to incorporate into our lives because it runs contrary to everything we know. Everything in our world from the corporate ladder to school to the military to sports to social gatherings to kid's karate classes is about "moving up", advancing, being the best we can be, being of higher rank, getting ahead of others around us. This is simply the way the world works. How high can you ascend in the pecking order?
But Paul tells us that this is the root of the Corinthians' problems. They lack unity because they strive after "more spiritual" leaders and they strive after those leaders because they misunderstand the role of Christian leadership within the Church and they misunderstand the role of leaders because they misunderstand the real meaning of spirituality in the Christian life. The root of all the problems Paul will address in this letter is that the Corinthians have failed to recognize that the gospel of the crucified messiah which they believed calls upon them to be a people of humility, sacrifice, service, and submission, a people whose wisdom is not of the world but is ordered and shaped by the cross of Christ.