well, it depends on the opening . . . and how aggressive both you and your opponent get . . . for instance, the very aggressive king's gambit gives up a pawn on the 2nd move. I've also played a game (vs the KID) where the only 2 pieces that left the board was a white pawn and a black pawn (I won by checkmate around move 20).
On average, how many turns into the game is the first piece captured?

2 or 3 moves after the rooks are connected, so averagely at move 15.
However, this answer, no matter how truthful it is, is meaningless. Some openings even require very early pawn exchanges.

A: None of the above. There is no "correct" or "one size fits all" answer here. Maybe these blogs will help you focus on more relevant questions to ask instead (in addition to providing you with some useful info, hopefully)
http://www.chess.com/blog/NimzoRoy/chess-opening-principles?_domain=old_blog_host&_parent=old_frontend_blog_view
http://www.chess.com/blog/NimzoRoy/endgame-faqs?_domain=old_blog_host&_parent=old_frontend_blog_view
Based on intermediate to advanced level games you've seen/played. I ask because I play against a.i. chess opponents and sometimes I feel I take too many turns setting up my defenses before making the first offensive move.