QGD Tarrasch
After 1.d4, best defense to learn for the beginner?

Depends on your style. If you like to be positional , play 1.d4 d5 and 2. .. c6 or 2. .. e6 and go classical. A more flexible move, I think , is 1.d4 Nf6 and play some wild Indian or Benoni. Both moves are sound and good.
The Budapest Gambit of course. Gambits like these are good for beginners because they improve tactical skill. Some of the general plans are simple as well.
Depends on your style. If you like to be positional , play 1.d4 d5 and 2. .. c6 or 2. .. e6 and go classical. A more flexible move, I think , is 1.d4 Nf6 and play some wild Indian or Benoni. Both moves are sound and good.
Indian defences and the benoni are not for beginners. Without basic knowledge of classical openings hypermodern ones aren't good.
And the budapest gambit is not a recommendable gambit.
Just play the queen's gambit declined, after that the slav, and after that you can already choose between classical and hypermodern openings.

I'll second the suggestions to play the QGD - it's the most straightforward and easy to understand for beginners. "Offbeat" openings like the Budapest can be occasionally used as weapons in a repertoire to surprise opponents but are not something to depend on or use as a primary opening. Here's how the opening might go in the QGD:

The Tarrasch is playable against:
- 1.d4
- 1.c4
- 1.Nf3
- 1.g3
- Queens Gambit
- Catalan
- Colle
- Torre
- London
- Veresov
"Offbeat" openings like the Budapest can be occasionally used as weapons in a repertoire to surprise opponents but are not something to depend on or use as a primary opening.
I use it as a primary opening and I'm winning games left and right with it, I just beat someone 150 points higher than me with it on lichess a few minutes ago as a matter of fact.
http://en.lichess.org/051hd1ut/black
That rook lift plan is winning me game after game, especially now that I'm learning the nuances of when I can use that plan. When they fianchetto the bishop on b2 it's a good indicator that it will work.
The Rubinstein variations aren't a problem either because I know how to play against white's dark squared bishop. Put your pawns on dark squares to shut it out of the game, meanwhile the pawns on dark squares blockade the opponent's pawns on white squares to shut the other bishop out, simple, and it works.
At club level the initiative counts for a lot, because it's only when people start to become very good at chess that they become good defenders, I think that's the last thing people learn.
Don't write bad things about the opening unless you've tried it, and by try it I mean actually play 100 games with it and read up on it so you know the themes and everything. It's a winning opening, I'd say I score about 75% wins with it.

QGD Lasker is a perfect opening for new players (beginners shouldn't bother about openings at all). Straightforward, easy to understand, little theory to memorize.
Later on Black may add more complex variations, like the Tartakower, or the highly theoretical Semi- Slavs.

Thanks everyone! The QGD it is!
Question: Many players at my level don't know to play 2.c4 after 1.d4. On move two, I usually see them playing 2.Nf3 or 2.Nc3. How do you suggest I proceed as Black after 1.d4 d5 2.Nf3, or 2.Nc3? What plan/strategy should I keep in mind?
Are the plans after 1.d4 d5 2.Nf3 much different than the plans after 1.e4 e5 2.Nf3?
As White, I always start with 1.e4, and with Black, I always follow 1.e4 with 1...e5, and am perfectly happy with both opening moves.

Thanks everyone! The QGD it is!
Question: Many players at my level don't know to play 2.c4 after 1.d4. On move two, I usually see them playing 2.Nf3 or 2.Nc3. How do you suggest I proceed as Black after 1.d4 d5 2.Nf3, or 2.Nc3? What plan/strategy should I keep in mind?
Are the plans after 1.d4 d5 2.Nf3 much different than the plans after 1.e4 e5 2.Nf3?
As White, I always start with 1.e4, and with Black, I always follow 1.e4 with 1...e5, and am perfectly happy with both opening moves.
2...Nf6

The Budapest Gambit of course. Gambits like these are good for beginners because they improve tactical skill. Some of the general plans are simple as well.
http://www.chessgames.com/perl/chessgame?gid=1500222
Thanks for yet another example, this game between Kramnik and Odesskij is similar to the one played between Lembit Oll and Romero Holmes, in both games white fianchettoed the bishop on b2 and grabbed the pawn on a5.
http://www.chessgames.com/perl/chessgame?gid=1503219
Whatever opening someone chooses I think you can't give up on it. If you stick with your opening, just like if you stick with chess, you'll get better at it. There's an incredible amount of nuance and detail in every opening, and there's no limit to how good you can get with one if you just keep using it and studying it. So to the OP, it doesn't really matter which opening you choose against 1.d4 what matters is that once you've chosen it, you stick with it.

You know. I'm an "intermediate" player whom cut his teeth on the QGD Tarrasch Defense as a 1.d4 defense and the more I think about it the better that suggestion sounds.
Beyond that study tactics and try not to drop peices.. and play long games, 3 days per move.. Hundreds of them.

Sorry, but "beginners should play the QGD to learn fundamentals" is just dogmatic crap IMO. I never replied with ...d5 to 1.d4 or 1...e5 to 1.e4 even as a beginner because I simply hated the resulting symmetrical positions. You have to play what you're comfortable with.
As a BEGINNER??? As a Beginner players learn how the horsey moves. Beginners don't hate symmetry - they don't even know what symmetry is. I call dogcrap.
You're being way too patronizing towards beginners. I distinctly remember after having taught me the rules the first thing my old man did was buy me a collection of annotated master games sorted by opening. As I played through these, even though I was obviously a total patzer I nonetheless quickly developed a feel for which opening setups were to my taste and which weren't.
With all do respect, I believe the initial dislike of symmetrical positions comes from beginners simply being unable to find moves that look like they do something. With openings like the KID or Benoni there are things like pawn breaks or pawn storms that at least give a player a" feeling of direction."
However when those players are forced down more classical lines such as 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.Nf3 O-O 6.Be2 e5 7.O-O Nc6 8.Re1 or 1.d4 Nf6 2.c4 c5 3.Nf3 where White refuses to play d5 because "it moves a piece twice in the opening" beginners with the Black pieces don't fare as well since there are no thematic pawn storms or traps for them to play.
It's probably the reason why things like the exchange French or Slav are so disliked as well.
I am looking for a defense to 1.d4 where the plans for both sides are fairly easy to grasp at the beginning level.