Around the time I was born, my parents were going to a Lutheran church. As such, I was baptized as an infant. In most cases in the Bible, baptism is connected with receiving the Holy Spirit. If so, then that may explain why I don't remember when I became a believer. However, there ARE verses such as those in Mark and Luke (below) that make a reference to believing as well, which I couldn't have done as a baby.
Mark 16:16 (NLT)
Anyone who believes and is baptized will be saved. But anyone who refuses to believe will be condemned.
Luke 3:3 (NLT)
Then John went from place to place on both sides of the Jordan River, preaching that people should be baptized to show that they had repented of their sins and turned to God to be forgiven.
As such, I am considering being baptized again as an adult. After all, I can now say that I believe and that I have repented, but there is little or nothing in the Bible about being baptized twice. Also, infant baptism doesn't seem to have become widespread until at least a century after Christ (Wikipedia).
I consider myself non-denominational on the side of Wesleyan, but I'll take answers from any viewpoint.