To preface: I'm still a rookie at chess, and part of the reason I joined this site is to improve my ability. This is a game I played today against the computer, and I think had a chance of winning. My notations are in the game. I started out confident at the opening and through the midgame, but blew it towards the end. I think my big problem was that I wasn't playing aggressively enough, but was focused on maintaining a good defense. Thanks for your thoughts/suggestions!

You can't win that behind in pieces without a checkmate sequence (especially with a computer?) The difficulties are already very evident at 5...e5 when you might play cxd5 and then later take the knight or remove yourself from the double attack at e4. Also, I would try to improve on the opening moves.

You were lost after 6 moves, wouldn't kick yourself for throwing away a win.

After you lose your queen the position is completely helpless, your annotations seem to suggest your moves afterwards were important, or 16. was the beginning of the end.

Why did you give up your bishop? Pointless sacrifice.

You should stop playing on hard mode.

Playing on hard mode and being challenged is better than having it given to you on a tray.  And all beginnings are hard, but what will help you the most is learning the principles of chess openings.  You can learn just a few of the principles without having to memorize opening moves, and principles will always stick with you if you practice them. 

Not when you're hanging pieces left and right.

How about the Chess Strategies post in the General discussion? (Although I think that the list can be improved upon.)

You can't win at the moment where Black have captured your queen, the Knight, and when you sacrifice the bishop with no reason? But that's a good game :)

Your first moves should be attempting to take control of the center. Don't castle so early.

Dude i dont want to bash you, but i really lolled when I read on move 16: "This was the beginning of the end ..."

You lost a piece on move 6 !!! Lost even a Queen on move 13 ,and you really think that you was starting to lose just on move 16????

After your "6.a4" the match was lost because you gave a light piece for nothing. I wouldn't even comment the opening. I suggest you to give a look on some opening  and generally on the open theory.

Thanks for the feedback and advice guys. I will take a look at and start improving on my opening moves!

Make sure you take in why that pawn fork worked. If your pieces are one file apart on the same rank they can be open for it!

Looking at your last game, I would say stay with the computer games. If you're White, open with the king or queen pawn by moving two ahead. If you're Black, copy the White's king or queen pawn moves. Then in your second move, it will often be that you'll want to bring out one of your knights, but not in a way that it will be attacked by a pawn in the following move!

If you use this advice, I can give you more.