In Acts 2:38, Peter tells those at Pentecost that you will be given the gift of the Holy Spirit after you repent and are baptized. Cornelius received the Holy Spirit as a sign to the Jewish Christians that Gentiles could be saved as well. There are only two instances where the Holy Spirit manifests itself as it did with Cornelius, the other being when the apostles spoke at Pentecost. However, in all other conversion accounts in Acts, including Cornelius, baptism is mentioned.
I have a question I would like for you to answer. Can you be saved before you have your sins removed?
Baptism does not remove your sins. Only the blood of Christ. We're getting into the multiple steps of regeneration. My belief is that you're saved the moment God wishes to give you a new heart. The rest are the results of God's work.