In regular chess, no. Are you doing odds or something?
Pawn first move & diagonal capture?

No. I am new to the game and playing my brother. He has had some experience years ago, and he said that he thought this move would be legal. I told him that from the books I have been reading it would not be. I told him I signed up on this site and would ask. So I believe that I would be correct in saying, the move I mentioned is an illegal move. Right?
Can a pawn on his first move, make his move, then at the same time, do a diagonal capture. In other words, the pawn moves let's say one square, and there happens to be an opponent’s piece at a diagonal after he made his move, can he make the move, and also then make the capture at the same time? So in essence, he would be actually making 2 moves at the same time.