Yes
How do you take chess pieces?

I do both. I find taking their piece off, then putting mine in the square their piece was occupying.

Blitz I have seen guys take off the piece with their off hand and move their own pieces to the square with their play hand and hit the clock. They mainly do it to save time on the clock. Me I usually take off the piece first and place my attacking piece in the square.

Blitz I have seen guys take off the piece with their off hand and move their own pieces to the square with their play hand and hit the clock. They mainly do it to save time on the clock. Me I usually take off the piece first and place my attacking piece in the square.
This is not legal i think. Everything must be done with one hand.

+1
This is what I do too - especially OTB blitz.

Great question here. With a plastic set I'll take my piece with thumb, index, and middle fingers, and smash it onto my opponent's piece with a nice crushing sound as I fluidly scoop his up with ring and pinky and palm it, leaving my piece remaining on the square.
I've seen plenty of even top players doing the remove the captured piece first method, but feel this is inferior for several reasons. First, it can get confusing if you're capturing a piece across the entire board. Second, if you take the piece off first then try to make the move and your opponent points out it's an illegal move, it can be confusing where that piece you were capturing originally was when you're trying to put it back. Whereas if you started to move your own piece to begin with it would be immediately obvious that it was an illegal move. Third, sometimes a piece can be captured several ways and taking the piece off the board then thinking which piece you want to use to capture it can also confuse matters. I'd be curious about the FIDE touch move rules on this topic.
Removing the piece first is maybe better suited for nice wooden sets. On a side note, procedure for how to Queen a Pawn is another topic to be explored.

Actually, neither. I do it all in one motion. It takes a certain amount of practice, I suppose (one of the defects of the movie The Defense was that the "players" were obviously unpracticed at capturing).
pushwood tells his niece "you're doing it all wrong!"

You gotta grab em like french fries--a whole handful at a time.
"i just grab 'em by the.......they let you do it if you're famous".

Blitz I have seen guys take off the piece with their off hand and move their own pieces to the square with their play hand and hit the clock. They mainly do it to save time on the clock. Me I usually take off the piece first and place my attacking piece in the square.
This is not legal i think. Everything must be done with one hand.
At club it is legal if the opponent doesn't mind. As long as it is at club and not a blitz tournament and both agree to it. Then what's the issue? To me play the game in club between opponents agreements. Tournaments play by tournament rules. That is how I look at it.

Blitz I have seen guys take off the piece with their off hand and move their own pieces to the square with their play hand and hit the clock. They mainly do it to save time on the clock. Me I usually take off the piece first and place my attacking piece in the square.
Isn't placing the capturing piece while taking the piece with the same hand in one motion faster?

Blitz I have seen guys take off the piece with their off hand and move their own pieces to the square with their play hand and hit the clock. They mainly do it to save time on the clock. Me I usually take off the piece first and place my attacking piece in the square.
Isn't placing the capturing piece while taking the piece with the same hand in one motion faster?
It is a bit faster I have noticed. I personally don't do the two hand capture. I have seen a couple of local players at the club do it when playing blitz.
Do you push the piece out of its square and then take it or first take the captured piece and replace it with your piece that captured it.