Most of the men involved in our church plant, Glory of Christ Baptist Church, meet on first and third Saturdays for a theological discussion. A couple of meetings ago the issue of the order of baptism and communion came up and I stated that I believe baptism should precede communion whenever possible. Later, one of the men wrote asking for further clarification and proof of my position and here is what I wrote to him. I welcome your input on this matter as well.
...About baptism and communion, here is the heart of my position:
1. There are zero instances in the Bible of a person taking communion before baptism, but there are several that display the pattern of repentance then baptism then entrance into body life, which includes communion (Acts 2:38-41, 8:12-13, 8:36-38, 9:18, 10:47-48, 16:15, 16:33, 18:8, 22:16). Given the number of texts that display this pattern, and the utter lack of texts that display the other pattern, I think the burden of proof is on those who would argue for communion before baptism.
2. First Corinthians 11:27-34 greatly elevates the nature of communion and the consequences of taking it unworthily:
 Whoever, therefore, eats the bread or drinks the cup of the Lord in an unworthy manner will be guilty of profaning the body and blood of the Lord.  Let a person examine himself, then, and so eat of the bread and drink of the cup.  For anyone who eats and drinks without discerning the body eats and drinks judgment on himself.  That is why many of you are weak and ill, and some have died.  But if we judged ourselves truly, we would not be judged.  But when we are judged by the Lord, we are disciplined so that we may not be condemned along with the world.  So then, my brothers, when you come together to eat, wait for one another—  if anyone is hungry, let him eat at home—so that when you come together it will not be for judgment. About the other things I will give directions when I come.
If a person is ready for this level of examination, reflection, self-assessment, and decision-making, and if a person passes the test, so to speak, so that he is reasonably confident that he will not incur the guilt of verses 27-28, then he is ready for baptism. And I believe, given the first point, that baptism should precede communion whenever possible.
3. I do not think that this pattern is absolute--that is, if a person receives communion before baptism he will not automatically incur judgment on himself. God is most concerned with the heart, and physical baptism is simply an outward sign of an inward reality. But I do think that this pattern is normative and I would question any person who claims to believe and is quick to take communion, but slow to be baptized. In other words, I would ask myself, "If they are willing to risk the judgment of 1 Cor. 11:27-28 but are unwilling to obey the command of Jesus to be baptized, are they truly saved? Are they uninformed? are they being disobedient?"
4. As all of this relates to children, I think the decision of when to allow children to take communion is ultimately in the hands of the parents. But allowing a child to receive communion is no small matter. Again I refer to 1 Cor. 11. And if a child is ready for this, then he is ready for baptism. If a child is not ready for baptism, surely he is not ready for the seriousness of 1 Cor. 11, that is, for communion.
I do think that there is room for differences of opinion here, but as Romans 14 states, the solution to our differences is not to take weak positions but for each of us "to be fully convinced in his own mind." And though I would not die on this hill, I am fully convinced in my own mind.
I welcome your feedback.