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 ...
Bringing out the queen in the opening is in general not the correct decision. The point is that usually the other side will often be able to develop their pieces to attack the queen and gain tempi.
However, if you still feel inclined to bring out the queen early, the only opening with a sound reputation I can recommend is the Scandinavian (
1. e4 d5) as black, and after the usual
2. exd5 Qxd5 your queen is out. Be prepared for
Nc3 and other moves that attack your queen, gaining your opponent a tempo, and allowing him to develop quickly.
If you play Black, Qxd5 Variation of the Scandinavian Defence is something that comes to mind...
[FEN ""] 1. e4 d5 2. exd5 Qxd5
Generally speaking, the queen likes open lines and lots of manoevering space, because she has long-range attack capabilities and being far away can't be targeted as easily by minor pieces. So if you enjoy playing the queen early despite all the dangers involved you should prefer openings that open lines quickly. Inbalanced positions with play on both sides of the board are preferable. Since the queen is an excellent attacking piece, look for gambits, too, because they usually aim at opening lines and sacrify material for speed of development, which is a must for a happy queen, since she usually needs support from at least another piece to work best. At example, look at the king's gambit or Latvian gambit.
The best way I can think of "free" my queen is to play an opening in which two or three pairs of minor pieces or pawns were quickly exchanged.
There is a danger of a queen being trapped on a "cluttered" board, and being forced to exchange herself for one or two minor pieces. Once a few of these are gone, the queen has a much freer scope over the board, in combination with rooks (occasionally with one of the remaining bishops).
One common opening playable against most mere mortals is the Scandanivian:
[FEN ""] 1. e4 d5 2. exd5 Qxd5 3. Nc3 Qa5
This brings the queen out early, but white often plays his bishop to d2 and later attacks the queen here (often by moving a later knight placed on Nc3, giving a discovered attack).
Regardless, this opening is playable in spite of the early queen sortie, not because of it. It's often used as an example of what not to do, but in this case isn't completely illogical.
The right answer is "don't do it," but the straight answer – which might be less helpful – is that
1 e4 e5 2 Qh5
is the best way for White to get the Queen out early, if White is really determined to do so. It's goofy but at least forcing. Best play continues
2 ... Nc6 3 Bc4 g6 4 Qf3 Nf6
White has already squandered the initiative by this point, though; Black now threatens
...Nd4 with a big gain of time.
5 Ne2 =/=
Another suggestion is to play the Evans Gambit, where White's
Qd1-b3 is often available. But unlike
2. Qh5, that depends on Black's response.
As others pointed out bringing the queen into play early is a double edged sword. That said, there are several main lines doing this.
Scandinavian was already mentioned. Then we have one Ponziani main line (although often the arguably weaker
3... Nf6 is played):
[FEN ""] 1. e4 e5 2. Nf3 Nc6 3. c3 d5 (3... Nf6) 4. Qa4
If you like to play aggressive, the Steinitz variation in the Scotch might be for you:
[FEN ""] 1. e4 e5 2. Nf3 Nc6 3. d4 exd4 4. Nxd4 Qh4
However be aware that you have a hard time if white doesn't cling to the pawn but attacks your weak spot c7 with the knight.
There are plenty of other choices (e.g. the Middle Gambit).