Besides Peter's sermon in Acts 2:38, does the Bible specifically say that we must be "saved" and/or receive "salvation?"
There are a lot of passages that talk about salvation, and the need to be saved. Jesus said that was his purpose. "For the Son of man is come to seek and to save that which was lost." (Luke 19:10) In fact, the name Jesus means "God saves." "And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins." (Matthew 1:21) One of the most familiar passages in the Bible says the purpose of Jesus coming to earth and dying was to save people. "For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. For God sent not his Son into the world to condemn the world; but that the world through him might be saved." (John 3:16-17) "Neither is there salvation in any other [than Jesus]: for there is none other name under heaven given among men, whereby we must be saved." (Acts 4:12)
In Acts 16:30-31 the jailer in Philippi assumed a need for salvation. See also Romans 5:10; 1 Corinthians 1:18-21; Ephesians 2:5-9; 1 Timothy 1:15; 1 Timothy 2:4; James 5:20; and 1 Peter 3:20-21. There are many more, but these should suffice to show that the Bible presents a need for salvation, and a savior.