Beginner post mortem...

I wanted to take a look, but your game is not posted using a standard format. Alot of others will probably skip it for the same reason. Here's how to post your games:
(1) The chess.com game board - When you post, the tool bar should display a brown & tan chessboard, a tree, a B, and I, and an A then "Font size". Click on the icon that looks like a chess board and it will step you through posting a chess diagram in a format easy for everyone to view.
(2) Alternatively, if the game was played on chess.com, post a link to the game.
(3) Alternatively, if the game was played on chess.com go to the "Moves" tab, click on View PGN, and use Edit | Cut and Edit | Paste to put it here.
You're right, 3.e3 was inaccurate. Correct is 3.cxd5 Nxd5 4.Nf3, and then you can follow up with 5.e4 to chase the knight away and gain a tempo. If black plays 4...Bf5 you can play 5.Nbd2 and follow up with 6.e4. But your move wasn't terrible, and it's certainly not why you lost.
5.Qa4+?! was pretty weak, as you said. It allowed black to develop a piece and took your queen out of the game. I don't like 6.a3?! either. Why were you trying to stop Bb4+? That move isn't very threatening: 6...Bb4+ 7.Bd2 Bxd2+ 8.Nbxd2, and black accomplishes nothing - other than helping white develop a piece.
8.g3?? was a big mistake. You needed to prevent Qxf2+. 8.Nf3 would work, so would 8.f3 or 8.Qc2.
14.Bd2?? was also a big mistake. Two black pieces are hanging here! You should have played 14.Ncxe2. You're already down a rook and two pawns and your position is already very resignable, but by playing 14.Bd2?? you go down an extra piece as well!
15.Kd3?? was bad too. Just take the knight: 15.exd4. At this point there's really no hope for you. You're down far too much material - time to resign and start over.
Sorry to sound like I'm ripping you up. In my analyses I like to call a spade a spade - if a move is bad, I say so. Maybe I should post some of my losses in here so that other people can rip me up.

Much better. :)
After 1.d4 d5 2.c4 Nf6 you could have played 3.cxd5! Now if 3...Nxd5 you have e4 and if 3...Qxd5 you have Nc3. In both cases you gain time and increase your opening edge.
Instead of 4.Nf3, you could have played 4.dxe5! and now Black's in trouble. Both his f6-knight and d5-pawn are under attack.
6.cxd5! is also very tricky for Black--his pinned c6-knight is under attack.
8.g3?? - You missed the attack on f2. 9...Bg4+ would've been disastrous.
10.Kc2?? - Nxf3 would've won the game on the spot. After missing this your game was in shambles and you never had a good opportunity to recover.
redearth329> "My opinion is that 3. Nc3 would have helped me out a lot more than 3. e3. I'm wondering if that's where it all went wrong or if that wasn't so bad and I botched it somewhere else, or if there were opportunities I missed to make up for it."
Studying openings gives you tiny advantages, like doubled pawns, the bishop pair, or in this case one extra tempo (move)... even a whole pawn in some cases!
You didn't lose because of move three, you lost because of tactical mistakes:
(1) On move 8, you missed that Black attacked f2 twice but you defended it only once.
(2) On move 12, You missed when Black hung his queen on f3.
This tactical area is called Counting. Here are some good links that discuss how to improve your tactical vision in that area:
Apr 2007: The Two Types of Counting Problems- 4th in the"Counting" series (A Counting Primer, The Most Important Tactic, and Is It Safe?)
I also highlighted several places where you could have handled your pawns better, but I would focus most of your energies on Counting tactics.