Pilgrim answered most of the exegetical issues. I will merely add,
Acts 2:38-41 Peter said unto them, Repent ye, and be baptized every one of you in the name of Jesus Christ unto the remission of your sins; and ye shall receive the gift of the Holy Spirit.
39 For to you is the promise, and to your children, and to all that are afar off, even as many as the Lord our God shall call unto him.
40 And with many other words he testified, and exhorted them, saying, Save yourselves from this crooked generation.
41 They then that received his word were baptized: and there were added unto them in that day about three thousand souls.
You have a good non-covenantal approach to the interpretation, unfortunately it is a covenantal set of verses. At least you understand the "congregation" addressed—the Jews, however, you failed to fully realize what the text means “from their perspective.”
The question is not if an infant can repent or trust in Christ (that is not in the text, nor should it be since it was common sense then, as it is today), but what a Jew in that day would have done about the sign and seal of his new found faith.
To be MORE correct this is not Peter's promise, but Peter was using a “biblical promise.”
Gen 17: 7 And I will establish my covenant between me and thee and thy seed after thee in their generations for an everlasting covenant, to be a God unto thee, and to thy seed after thee.
Compare the two sets of verses—see their similarity? Acts 2:38-39 associates circumcision and baptism!
First, in vs 38, the Apostle Peter calls for (1) repentance, (2) faith in Christ, and (3) baptism
by Jews (of course others were present). The clear link is seen in v. 39 where Peter gives the reason for this action: "the promise is to you and to your children, and all who are far off...." (compare with “and thy seed after thee in their generations,” etc.—read Gen 17). The Apostle Peter consciously uses the same formula (because it is an
everlasting covenant) in his preaching as the LORD himself used when he instituted the sign of circumcision in Genesis 17. The Jews listening understood this precisely. Thus, they understood as circumcision was for Abraham and his seed, thus was baptism for those that believe and their seed. Thus, the link is established with Peter, however not ONLY with him.
As Scott Clark states, the connection between baptism and circumcision is quite clear in Colossians 2:11-12. The connection is not direct, but indirect and the point of contact between them is Christ and baptism is the sign and seal of that circumcision. In v.11 Paul says "in him [i.e. in Christ] you were also circumcised with the circumcision done by Christ" and in v.12 he says exactly how it is that we were circumcised in and by Christ: "having been buried with him in baptism and raised with him through your faith...." For Paul, in the New covenant, our union with Christ is our circumcision. In baptism, we are identified with Christ's baptism/circumcision, as it were, on the cross.
The point not to be missed here is that, in Paul's mind, baptism and circumcision are both signs and seals of Christ's baptism/circumcision on the cross for us. By faith, we are united to Christ's circumcision and by union with Christ we become participants in his circumcision/baptism. Because circumcision pointed forward to Christ's death and baptism looks back to Christ's death, they are closely linked in Paul's mind and almost interchangeable. Paul's point here is to teach us about our union with Christ, but along the way we see how he thinks about baptism and circumcision and his thinking should inform ours.
