What is the best way to free your queen as soon as possible? We can open games in many ways (any of the pawns or knights) and then go ahead with different strategies. The queen being the most important piece, I would like to get it out in the field without compromising the security of the king and using a minimum number of moves. Which types of opening and/or moves would facilitate this? Also provide any pros and cons to your strategy.
I don't think that getting your queen out early should be your goal in the opening.  Fighting for the center, developing your pieces, and king safety should be your priority.  That said, e-pawn openings are your best bet to open lines for your queen as soon as possible.  Look into the Ruy Lopez:

[FEN ""]
1. e4 e5
2. Nf3 Nc6
3. Bb5 ...