You can do it either way. I prefer touching my piece first and removing the captured piece as part of the move, in a similar manner to the way you describe doing it.
capturing pieces OTB

Thanks Alvin.
I was wondering the same things recently.
Here are a couple related things I was wondering:
(1) Is there some protocol if you accidentally knock your king over when reaching for your piece? (I do this a lot because I often play in a jacket or sweater. I don't want my opponent to think I resigned!)
(2) Is there anything wrong with setting a queen down on the square where your pawn will be queening, without having to actually move the pawn to that square first? (I like to already be holding a queen in my hand when I'm about to queen--I figure it's rather intimidating!)

sqod,
In OTB rules, there must be a deliberate touch of the piece in order for you to be required to move it. I asked somebody this at an OTB tournament I was at a while back and he said, "Yeah, if you accidentally knock over a piece trying to reach for another piece, I don't think any one's actually going to require you to move it." I've read that if you accidentally knock over your king, that is not resigning either. There was once a game apparently where the guy knocked over his king accidentally but placed it on the wrong square and neither player realized it. But he wasn't required to move the king or resign.

1. Just say "J'adoube" or "I adjust" and put the piece back where it was.
2. You can remove the pawn or put the new piece down in any order you like. The pawn doesn't have to touch the promoting square.

uscftigerprowl,
Sorry, we were talking about touch move rules and deliberately touching a piece vs. accidentally knocking over a piece.

In chess, there is such a rule as "touch move". So there's some issue if you touched your opponent's pieces. In our local club, here's the rule:
1. Touch your piece and you are bound to move it to any legal square. If the touched piece do not have a legal move, you must move your King to any legal square.
2. Touch your opponent's piece and you are obligated to capture it. If the opponent's piece cannot be capture, you must move your King to any legal square.
3. Touch any square on the board and you are obligated to move any piece on that square. If no piece can move to the square, you must move your King to any legal square.
4. If you are obligated to move your King in either #1, #2 or #3 above, yet your King does not have a legal move, you lose the game.
The official rule, for the USCF at least, is that if the touched piece has no legal move or there is no legal capture for the touched opponent piece, then any piece can be moved.
There are no rules about touching squares/the board that I'm aware of, especially in regards to piece movement.

uscftigerprowl,
Oh, I understand what you're trying to say now. Thanks for the clarification. I did watch the video. It was too fast to see how they captured pieces though.

I have seen both, and I prefer to use my hand to move my piece to the desired square, and just then I pick up the taken piece

In chess, there is such a rule as "touch move". So there's some issue if you touched your opponent's pieces. In our local club, here's the rule:
1. Touch your piece and you are bound to move it to any legal square. If the touched piece do not have a legal move, you must move your King to any legal square.
2. Touch your opponent's piece and you are obligated to capture it. If the opponent's piece cannot be capture, you must move your King to any legal square.
3. Touch any square on the board and you are obligated to move any piece on that square. If no piece can move to the square, you must move your King to any legal square.
4. If you are obligated to move your King in either #1, #2 or #3 above, yet your King does not have a legal move, you lose the game.
The official rule, for the USCF at least, is that if the touched piece has no legal move or there is no legal capture for the touched opponent piece, then any piece can be moved.
There are no rules about touching squares/the board that I'm aware of, especially in regards to piece movement.
I've noticed over the years that casual chess clubs sometimes follow nonstandard rules, like you allude to in Alvin's post. Sometimes club members know that these nonstandard rules are not official USCF/FIDE rules, but sometimes they mistakenly believe that their club rules are official.
So when visiting a new club, I ask if they follow official rules, and bring my rule book in case I need to point out misinterpretations. Needless to say, some of these club members don't take kindly to a newcomer telling them they've been doing things incorrectly.

The chess club I go to on occasion advertises itself as very casual. Some play with clocks, some without. It's about improving chess rather than competing in chess.

In our club, we are very casual during normal club meetings. Most of the time we don't use clocks, unless we want to get a game in but only have limited time (just to make sure the game doesn't last too long). Touch move is not enforced and take-backs are usually allowed; though if too many happen it is usually not allowed for the rest of a particular game.

The chess club I go to on occasion advertises itself as very casual. Some play with clocks, some without. It's about improving chess rather than competing in chess.
That's a very good approach for keeping a healthy number of members around -- keep it relaxed and supportive. If the competitive attitude gets too intense, not only does it scare new people away, but it can even become offensive to long-time members who are there for the comraderie.
I have seen clubs fall apart because of a strong new member who is too wrapped up in making sure everyone knows he is the alpha-male (not being sexist, I've just never met a female who behaved this way). The other type to watch out for is the one who is not that strong, but takes every defeat as a blow to their maladjusted self-image. I've seen this latter type too. I nearly had to call the police because he physically threatened other players who he lost to. Anti-social individuals need to be confronted early on so that they don't damage the club. But this must be done discretely and gently, so that they have a chance to save face. Fortunately, in my experience, 95% of people are courteous and reasonable.
EDIT: sorry I went off on a tangent there, but these issues all seem inter-related (arguments about rules by hyper-competitive players, etc).

Either way is fine, I, for one, do a hybrid, depending on where the piece is. If I'm taking something from across the board, I'll often just remove it first, to minimize my chances of fumbling the piece. In general, though, I'll take as part of the move.

Anyway, I came across something when I was a child. Out of topic though.
Castling, is it acceptable to move the Rook first and then the King?
Or is it a rule to move the King first and then the Rook.
Reading the description of castling in any beginner book, it should be King first. I will not argue about computer chess because they are programmed to be King first. This can only be an issue in OTB, just lie capturing pieces.
Always King first, move it over two squares, then the rook to the opposite side of the king. 100% required under FIDE rules and under USCF it is better to do it that way to avoid any confusion. I don't have the rule book in front of me right now but I believe it comes down to intent and if you can avoid any judgement calls by moving the King first. Also, it is supposed to be done with one hand.
As to your other reply to me, I understood that it was a club rule. Also, annoying your opponent is against the rules and constantly touching squares on the board could fall under that.

Castling is typically considered a king-move as it is the only time the king can move more than one square. Just get into the habit of moving the king first.

I was playing correspondence chess with some friends over the Spring and when I was looking at my chess board, I would always capture my opponent's pieces by touching my piece first and then contacting their piece (perhaps with it or just with my other fingers) and removing it from the board. Of course, touch move doesn't apply in correspondence chess but I liked to have the board set up so I could see what was going on any way.
Then, I go to a chess club that meets at a bakery and I play a couple people who instead of touching their piece first when capturing my piece, they touch my piece first, remove it from the board, and then place their piece where mine was.
So here is my question--is there any "right" way to capture a piece? Should I be touching my opponent's pieces first?
kinky way you play chess! i like this way of thinking
I'd say always try yo get opponent to touch your piece instead of touching his, i like passive pleasure
I was playing correspondence chess with some friends over the Spring and when I was looking at my chess board, I would always capture my opponent's pieces by touching my piece first and then contacting their piece (perhaps with it or just with my other fingers) and removing it from the board. Of course, touch move doesn't apply in correspondence chess but I liked to have the board set up so I could see what was going on any way.
Then, I go to a chess club that meets at a bakery and I play a couple people who instead of touching their piece first when capturing my piece, they touch my piece first, remove it from the board, and then place their piece where mine was.
So here is my question--is there any "right" way to capture a piece? Should I be touching my opponent's pieces first?