Amy G.: You are spot on.
Peter had a big long discourse I guess he intended to get into. That is what he implies at "as I began to speak" (NKJV) in Acts 11:15.
Now, back to Acts 10, we see that the Lord had other plans.
Act 10:34 And Peter opened his mouth and said, Of a truth I perceive that God is no respecter of persons:
Act 10:35 but in every nation he that feareth him, and worketh righteousness, is acceptable to him.
Act 10:36 The word which he sent unto the children of Israel, preaching good tidings of peace by Jesus Christ (He is Lord of all.) -
Act 10:37 that saying ye yourselves know, which was published throughout all Judaea, beginning from Galilee, after the baptism which John preached;
Act 10:38 even Jesus of Nazareth, how God anointed him with the Holy Spirit and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him.
Act 10:39 And we are witnesses of all things which he did both in the country of the Jews, and in Jerusalem; whom also they slew, hanging him on a tree.
Act 10:40 Him God raised up the third day, and gave him to be made manifest,
Act 10:41 not to all the people, but unto witnesses that were chosen before of God, even to us, who ate and drank with him after he rose from the dead.
Act 10:42 And he charged us to preach unto the people, and to testify that this is he who is ordained of God to be the Judge of the living and the dead.
Act 10:43 To him bear all the prophets witness, that through his name every one that believeth on him |receives| remission of sins.
Act 10:44 While Peter yet spake these words, the Holy Spirit fell on all them that |were listening to| the word.
Act 10:45 And they of the circumcision that believed were amazed, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Spirit.
Act 10:46 For they heard them speak with tongues, and magnify God. Then answered Peter,
Act 10:47 Can any man forbid the water, that these should not be baptized, who have received the Holy Spirit as well as we?
Act 10:48 And he commanded them to be baptized in the name of Jesus Christ. Then | they asked him to stay on for a few days. (ASV|ESV|ASV|NASB|ASV|NASB)
Here is what stands out to me:
Act 10:43 To him bear all the prophets witness, that through his name
every one that believeth on him |receives| remission of sins.
Act 10:44
While Peter yet spake these words, the Holy Spirit fell on all them that |were listening to| the word.
First, Peter made clear regarding Jesus Christ that "everyone that believeth on him receives remission of sins." His audience was listening receptively, and so when this was said, they believed, and God sent the Holy Spirit.
In Acts 15:8-9, Peter reported about this “And God, who knoweth the heart, |gave them evidence|, giving them the Holy Spirit, even as he did unto us; and he made no distinction between us and them, cleansing their hearts by faith” (ASV|NBV|ASV). When they believed at Acts 10:43-4, God sent the Holy Spirit, cleansed the hearts of the believers for their faith, and testified to the unexpecting Jews that these were saved believers.
Further, it was after Peter saw that these people were now Christians that he called for baptism. In Matthew 28:19-20a Jesus Christ said “make disciples of all the nations|. Baptize them in the name of the Father and the Son and the Holy Spirit. Teach them to obey everything that I have taught you” (NASB|NCV). Per Acts 11:26, “Christians” is a rename of “disciples.” Jesus Christ said regarding "disciples" that we should "baptize them" meaning that we baptize Christians -- not unbelievers. Peter followed this; when he realized that these people were now Christians, he called for their prompt baptism.
In bringing up Acts 10 again, you were spot on. Acts 10 has actually been brought up earlier, but it was a while back. Acts 10:43 makes clear regarding Jesus Christ that "everyone that believeth on him receives remission of sins" as quoted above. There are people who wrongly teach that if someone who "believeth on him" fails to get properly baptized, s/he will not get remission of sins. However, according to Scripture, there is no way that someone who "believeth on him" does not receive "remission of sins."