[Winman]

And what did Peter preach in Acts 10?

Acts 10:34 Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons:
35 But in every nation he that feareth him, and worketh righteousness, is accepted with him.
36 The word which God sent unto the children of Israel, preaching peace by Jesus Christ: (he is Lord of all: )
37 That word, I say, ye know, which was published throughout all Judaea, and began from Galilee, after the baptism which John preached;
38 How God anointed Jesus of Nazareth with the Holy Ghost and with power: who went about doing good, and healing all that were oppressed of the devil; for God was with him.
39 And we are witnesses of all things which he did both in the land of the Jews, and in Jerusalem; whom they slew and hanged on a tree:
40 Him God raised up the third day, and shewed him openly;
41 Not to all the people, but unto witnesses chosen before of God, even to us, who did eat and drink with him after he rose from the dead.
42 And he commanded us to preach unto the people, and to testify that it is he which was ordained of God to be the Judge of quick and dead.
43 To him give all the prophets witness, that through his name whosoever believeth in him shall receive remission of sins.
44 While Peter yet spake these words, the Holy Ghost fell on all them which heard the word.
45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
46 For they heard them speak with tongues, and magnify God. Then answered Peter,
47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.

1) The word was sent unto the children of Israel
2) This word began in Galilee after the baptism which John preached
3)They were commanded to teach Jesus was the Judge of the quick and dead
4) All the prophets gave witness through believeing on Jesus they would receive the remission of sins

Note that the these Gentiles received the Holy Ghost upon hearing the word of God and believeing. And that the Jews were astonished because they received the Holy Ghost "as well as we?"

I can't be dogmatic about this, but I believe this is a strong argument that the Jews in early Acts received the Holy Ghost by hearing and believeing the word of God, and were baptized afterward, just as these Gentiles.

But it is clear that the message was the forgiveness of sins upon believeing on Jesus, not the restoration of the kingdom.