Romans 1:16a
"For I am not ashamed of the gospel of Christ: for it (the Gospel) is the power of God unto salvation to every on that believeth;..." Another good one is found in I Corinthians 1:18
I Corinthians 15:1-4
"For the preaching of the cross is to them that perish foolishness; but unto us which are saved it is the power of God. (unto salvation.)
And how do we obtain salvation as we just saw in Romans 1:16? By believing! Not repenting as Peter preached. And this is Paul's theme all the way through. Some more you can look at are found in I Corinthians 15:1-4. Romans 10:9-10, I Thessalonians 4:14, and there and many others in Paul's epistles, he never changes that theme. Now let's come back to the Philippian jailer and see that his family was also involved. It wasn't just a one man deal, so we pick that up in verse 31. Paul is speaking here.
Acts 16:31
"And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, (now these next three words throw a curve at a lot of people) and thy house."