The bishop's are unique chess pieces. They are the overseers of the chessboard. In the opening, four pawns can directly free both bishops, yet two could do.
By losing a bishop, we weaken a color and the pawn structure becomes complex for the rest of the game.
The chessboard only has two colors, and by losing a bishop we have one less piece to protect or attack a particular colored square. If the opponent has more pieces to attack a weak colored square, then they can probably gain an advantage from that.
Maybe this is one of the reasons why two bishops are worth more than two knights. Are there more reason why two bishops are worth more than two knights?
Which bishop do you tend to capture first and why?
The black one because i'm racist :)
Think of the two colors as recolored shadows. There are dark colored chess pieces and light colored pieces. All the chess pieces have a role to play. The role of the bishops are overseers.
Which bishop do you need to protect? Is it the King's bishop because you castled king side or is it the Queen's bishop because you castled Queen side?
I tend to hold on to my king's bishop the most because, I tend to castle King side. So I assume it is the one my opponent would want to capture to weaken my castled king.
Well, it depends. That's the best answer you can get.
If you have no light squared bishop but have a lot of white squares behind or around your pawns, then you might have problems because you can't pretect it.
As such, it's not a matter of King's or Queen's bishop. If you fianchetto your King's bishop, then you've exposed squares on f3, h3 and even g2. Your bishop, however, is protecting those squares so don't lose it!
Imagine a normal English set-up with c4, d3, Nc3, g3 and Bg2. You'll notice that all of white's pieces are controlling white squares, particularly the long diagonal. With that, your King's bishop also is important but if you flipped the position, fianchettoing your Queen's bishop, then that bishop would be more important as you are controlling dark squares.
Another example: the Stonewall. The Stonewall with d5, f5, e6 and c6 leaves a lot of weak dark squares. As such, your dark square bishop is really important. (Same if white does the stone wall)
Your bishop, just because it is your king's bishop, won't necessarily stay on the side it starts on. Often, it doesn't. Eg. Ruy Lopez. 1. e4 e5 2. Nf3 Nc6 3. Bb5. Already your bishop is on the other side of the board and has nothing to do with your king. Often, it might end up on b3 having NOTHING to do with your castled king at all.
Last point: endgames. Opposite-colour bishop endgames are often drawn, even if one side has more pawns (even up to 3!). Additionally, you normally want your pawns on the opposite coloured squares to your bishop. This, *might* affect what you do with your, or your opponent's, bishops. Like trading off the 'bad bishop' in the French. (As a side note, I personally feel the bishop isn't a problem in the French. It will come out.)
So with that, it just depends. Position by position. I would not trade a strong bishop for a developed one, for example, without a very good reason. No matter the colour.
