I've recently become enamored with the idea that in the opening, both sides should strive to get all their pieces off the back rank so that the Rooks are connected. For each side, this requires a minimum of 8 moves: 2 pawn moves, 2 Knight moves, 2 Bishop moves, a Queen move, and a castle.

I also think that the best chess moves are those that develop, attack another pawn/piece, defend one of your pawns/pieces, and prevent the opponent from making a similar move.

Are there any resources on Chess.com that might explain the theory behind certain openings? For example, as Black I'm thinking about something like 1. e4 d5 2. exd5 Nf6 3. c4 e5 4. dxe6 Bxe6 and here Black is 4 moves from connecting his Rooks and White is still 7 moves away. White is up a pawn, but I feel like Black's got to be able to recover the pawn at some point, and if not, it looks like Black is controlling the game (if the d pawn ever goes to d4, Black will recover his pawn, otherwise, it goes to d3 and hems in the light squared Bishop). 

It's not a bad idea to begin playing competitively with gambits and other lines which emphasize quick development.  As tonydal notes, it's no panacea, but it ain't half bad.

It's definitely a good thing to develop one's pieces as quickly as possible.  The trick is figuring out which squares are best for each, remembering they need to be able to cooperate as well as perform individually.  Since this is rarely apparent for all pieces so early, in most openings a few pawn moves will be interspersed, creating opportunities for one's pieces and restricting the opponent's to determine how one's forces are optimally deployed.  Quickly developed pieces can't help if they are on the wrong squares.

If you achieve developing all your pieces and connecting your rooks in all your games, I have no doubt you will win a majority of your games.