Alexsmum, the thing about confirmation is, you can only get it done in the same denomination that 'baptized' you as an infant. The Baptist church has no concept of confirmation, for example, because it only baptizes those who confess faith for themselves.
This all begs the question, if confirmation allows an adult to own for themselves, vows made on their behalf when they were an infant, what's the point baptising the infant in the first place... ? There certainly is no scriptural imperative to do it; the Bible often commands that people believe and be baptised and never explicitly states that children are, or ought to be, baptised.
There is an interesting study on the subject to be found here which makes a very good point about one of those 'household' baptisms in Acts ... At Acts 11:14, the entire household of Cornelius is baptised. This is often used as an argument in favour of infant baptism, as 'entire' would logically mean any children present would also have been baptised. Of course, the passage itself does not make clear either way (nor do any of the other household baptism references in the NT). However, Acts 10:44-48 gives us further information about the conversion of Cornelius and his household. It says that everyone who heard, received the Holy Spirit. If there were infants in the household, as some argue, then the meaning of this passage demands we believe that they, too, listened to Peter preaching, believed him, and received the Spirit. I suggest, most humbly, that such a scenario is daft. There clearly were no infants involved in the baptism of the entire household of Cornelius.