Babies in Church
I like how the NLT phrases 3:2-3:
"I had to feed you with milk and not with solid food, because you couldn’t handle anything stronger. And you still aren’t ready, for you are still controlled by your own sinful desires. You are jealous of one another and quarrel with each other. Doesn’t that prove you are controlled by your own desires? You are acting like people who don’t belong to the Lord."
This passage isn't necessarily a very famous or popular couple of verses - you don't see it on pleasant Christian paintings or hear that it's someone's "life verse" - but it is hugely important. Along with a lot of other passages, these verses show a clear connection between what we know about the Bible/theology and our action.
Paul wanted to teach them so much more than he did. He wanted to teach them to know God deeply and understand the Scriptures well, but he couldn't. In essence he says here, "I wanted to treat you like big people and teach you things that are important and worthwhile, but you're not ready for it. Instead I had to treat you like babies." He goes on to say that they're still acting like babies! Why? Because they are controlled by their sinful desires. Because they are jealous. Because they fight. The last sentence sums it up: "You are acting like people who don't know God."
Wow. Well, Paul isn't really known for holding his feelings back, and this is definitely another case of brutal honesty. There's a reason for his honesty: this isn't just a case of spiritual babies wrecking their own lives: "By remaining immature and allowing that immaturity to divide them, they were wreaking havoc on the church."  Also, "Their quarreling not only mocks the cross-centered gospel (1:18–2:5) but fails to recognize the fundamental equality of all believers when measured against the attributes and character of God." 
In this case Paul is speaking specifically of those who claim a certain "famous" person as their master, making them better than someone else. This would probably be akin to saying "I follow Pastor Greg" or "I follow Pastor ________" in a church context. Let's clarify here: we all follow JESUS. This is what Paul focuses on here, but his comments can definitely be extended to division and quarreling under other circumstances as well.
So what does this mean for us today? What's the application?
"The foundation of the church—of all believers—is Jesus Christ. Nothing and no one else will do, wrote Paul. A building with no foundation, or one poorly constructed, will not last. The finest materials used to construct a home quickly rot and fall apart if they are resting on the ground. And a building is only as solid as its foundation. The foundation of our lives is Jesus Christ; he is our base, our reason for being. Everything we are and do must fit into the pattern provided by him. Are you building your life on the only real and lasting foundation, or are you building on a faulty foundation such as wealth, security, success, or fame? Be careful how you build." 
A shared national culture or socioeconomic culture or age culture will NEVER be enough to hold a church together. You can hold things together just by having a similar background - a church holds together through the foundation of Jesus. If He is foundation, He will always be there through the fighting and confusion and frustration. He MUST be the foundation for a church to last.
Also, think about yourself: what would Paul say to you? Are you eating spiritual "solid food"? Are you understanding the Bible deeply and being changed by it? If not, why? Is your life controlled by your desires rather than the Spirit? Are a life indistinguishable from someone who doesn't know Jesus? If you want to get closer to God but feel frustrated, are you doing the basic "milk" of spiritual life? Are you studying the Bible regularly? Are you praying regularly? Are you examining your life and repenting of your sin regularly? If you're not, I think I can tell you right now why you feel far from God.
This passage is rich with implications for our churches and spiritual lives. Don't keep it in the past: thinking this is just about the messed up church in Corinth with its baby Christians. Make sure you ask yourself: "Is this us (as a church)? Is this me?" If it is, pay VERY close attention to what Paul says in this passage. Take his warning. Do you what you can to fix it: cling to the cross.
 Craig Blomberg, 1 Corinthians, The NIV Application Commentary (Grand Rapids, MI: Zondervan Publishing House, 1994), 72.
 Barton and Osborne, 52.