How to handle bishop after bg5 in queen's gambit declined?

Hi.

I'm an absolute beginner at chess, I started less than 3 weeks ago from the basics (like I even had to check how pieces move), so sorry if the question seems obvious to you.

Let's assume I'm white, and I want to play queen's gambit. Black declines the gambit, so it becomes queen's gambit declined.

1. d4 d5
2. c4 e6
3. nc3 nf6
4. bg5 h6

Now, what happened is that obviously I pinned the knight with the bishop, and as expected black used his pawn in order to make white's bishop retreat. But what now?

White can choose to either trade bishop for knight with bxf6, retreat with bh4 or go to [f4|d3|e2|c1].

Which is the best move, and why? I usually retreat to h4 because the guide I'm following suggests that. Is one of the other options usually better, the same or worse?

Now, assuming I do retreat to h4, what if black goes to g5 (so 5. bh4 g5)? Is that a good move for black? Obviously white will have to save bishop with bg3, but doesn't that leave the bishop in a bad position? On the other hand, black has destroyed his pawn formation on the king's side, so that's a good thing for white I think.

tldr; after the four moves I wrote above, what are the best moves for both black and white? Which is the worst possible move?

Thanks to whoever decide to answer :). If something is unclear, poorly formatted or goes against some rule, please tell me so.

"White can choose to either trade bishop for knight with Bxf6, retreat with Bh4 or go to [f4|e3|d2|c1].
Which is the best move, and why?"
The best move is presumably 5 Bh4.
5 Bxf6 loses the bishop's pair and gives your opponent an unopposed bishop of the dark squares. It has been played however. Black recaptures either with his queen 5...Qxf6 or with a pawn 5...gxf6.
Retreats to f4, e3, d2 or even c1 mean that you have given your opponent a free move 4...h6, as you could have played your bishop there immediately instead of to g5 first. For example 4 Bf4 is a decent move, but 4 Bg5 h6 5 Bf4 is not as good.

"Now, assuming I do retreat to h4, what if black goes to g5 (so 5. Bh4 g5)? Is that a good move for black? Obviously white will have to save bishop with Bg3, but doesn't that leave the bishop in a bad position? On the other hand, black has destroyed his pawn formation on the king's side, so that's a good thing for white I think."
You are right: on 5...g5 6 Bg3 and black has weakened his pawn structure. The bishop is happy on g3: it has a nice diagonal h2-b8 that runs through the central square e5. Black can now play 6...Ne4, or 6...c5, or 6...Bb4.

"what are the best moves for both black and white? Which is the worst possible move?"
Nobody knows for sure. Presumably either 5 Bh4 Be7 or 5 Bh4 Nbd7 or 5 Bh4 c6. There are no absolute best moves: a lot is fashion. What is true today is wrong tomorrow. You do not need to know: just apply logic and play your own moves and you will be fine. Variations are transient, method is eternal.

Here is a top game where white takes 5 Bxf6 Qxf6 6 Nf3. There is a slight transposition of move order, but it leads to the same position.

Thank you very much. Clear and helpful answer :)

Actually after 1. d4 d5 2. c4 e6 3. nc3 nf6 4. Bg5, 4..h6? loses a pawn or black has doubled pawns after 5. Bxf6!, 5..Qxf6 just loses the d5 pawn for nothing, and 5..gxf6 gives black ugly doubled pawns. So your original question is flawed in that black never plays ..h6?. It needs preparation and there are many lines that you don't have to worry about h6 at all.

The actual valid moves, in the order of popularity, are 4.. Be7 (favorite by far), 4..Nbd7, 4..c6, and then finally 4..Bb4 which white usually play 5. Nf3 for the Ragozin variation:

http://www.chessgames.com/perl/chessopening?eco=d38

The Ragozin is known in general to be good for white. And I think only Bb4 would bring the question of where the bishop should go? If the black bishop is not going to b4 then h6 Bh4 looks very normal to me.

I recommend instead of 4. Bg5 immediately, to first play 4. cxd4 (I do this 100% of the time) and after exd5, then you play 5. Bg5. You get the same lines but avoiding any later dxc4, which black may actually win a pawn and keep the annoying pawn on c4 if you are not careful.

4. Nf3 is also a good move. The difference is while Nf3 is necessary and almost forced (you are not playing it to Ne2 or Nh3), Bg5 is not necessarily the case. Depending on black's 4th move it is conceivable that you may develop the bishop to f4, keep it on defense at d2, or start preparing for 0-0 instead. And after a later h6 Bh5 the bishop may be off the a5-e1 diagonal earlier than you want.

After 4. Nf3, black usually declines the Ragozin by playing the more common 4.. Be7 (which 5. Bf4 makes a lot of sense and scores better in general), or 4.. c6 which 5. Bg5 makes sense.

The best thing the OP can do at his level is stop worrying about opening theory and work on tactics + endgames.

Agree with #5.
On #4: 4...h6 does get played. Indeed move order matters: after 3 Nc3 instead of 3 Nf3 black loses pawn d5 if he recaptures 5 Bxf6 Qxf6. However, the recapture 5...gxf6 sees play. The black pawn structure is shattered, but black has the bishop's pair and open files. Example;

Just to add two thoughts.
You should study opening theory enough to get you to a playable middlegame. By asking these questions about the opening, you are studying opening theory, but don't take it to the other extreme where you have no idea about how to continue.
After the normal moves, 4...Be7 5. e3 (instead of the immediate h6), Karpov would exchange, ever if the bishop wasn't attacked, to help promote his minority attack. Whereas I prefer the bihsop on f4, controlling b8, when I do a minority attack.
So the best bishop move depends on which positions you feel comfortable with. If you prefer bishops, don't exchange even if the (former) champion does. If you prefer a pawn storm, exchange the bishop, castle queenside, and throw those pawns forward.