It doesn't matter. Look at the variety of opening moves for black and white in a book on chess openings, or google "chess openings" to see the best options to start a game of chess:)
Pawn or Knight?

usually in layman's terms it is always better to move the pawn as they open up the path to other pieces

If you are playing white, d4 is the best move while e4 is slightly more played. Both are good. However, Nf3 scores better for white than e4, so do e4 and g3. So it's up to your taste that which move should you play. But generally, pawn moves are preferred.
If you're black instead, there are a variety of different options. Nf6 and d5 are good options against d4. Nf6 is somewhat better and popular. Against e4, very few people knight moves. Pawn moves like c5, e5, e6 and c6 are played a lot.
Now it's your task to decide which openings suit you. Good Luck!

Is it best to move the Knight on the opening move, or a pawn on the opening move?
Does it depend if your playing black or white???
What you play in the opening matter's. Especially with the white piece's.
If you wish to know what the best move is at move 1 unbiasedly I will show you factual evidence of each move.
Here are visual example's.
1.e4 allows 16 squares to be controlled.
1.d4 allows 14 squares to be controlled.
1.c4 allows 12 squares to be controlled.
1.Nf3 allows 11 squares to be controlled.
1.Nc3 allows 11 squares to be controlled.
Here is your question: Is it best to move the Knight on the opening move, or a pawn on the opening move?
1.e4 allows 16 squares to be controlled.
1.d4 allows 14 squares to be controlled.
1.c4 allows 12 squares to be controlled.
1.Nf3 allows 11 squares to be controlled.
1.Nc3 allows 11 squares to be controlled.
The text highlighted in green is the pawn moves the text highlighted in blue is the knight move's. So the answer to your question is the Pawn moves are better factually becuase they control the most squares.
Among the pawn moves the best pawn move that controls the most squares is e4. This claim I am making is a factual claim. The move 1.e4 is the most active developing move in the starting position of the game. It activates the most squares in effect as you can visiual see.
However, other moves can still be played that have been tested by chess players world wide. So even though 1.e4 is considered the most active best move it is certainly not the only playable move.
All the above moves I have shown are playable and have been played before by Title players through out history.
Among the knight moves the best knight move is considered to be Nf3 the reason why is becuase both knight moves control the same number of squares;however, since the white kings starting position is on the right side of the board Nf3 is considered better than Nc3 becuase it allows the white king to castle faster.
So that is the answer to your question about the best opening move's.
To answer the second question: Does it depend if your playing black or white???
White moves first so white does not have to depend on anything. The white sided player can play the move they wish to play whether it is the most active or not.
As for Black....They are doomed by their limitation of playing second. So their move depends completely upon what white moves. They can still try to play the most active move's;however, it will change depending on what white play's.
Hope that helps. Have a nice day.

I would like to show 2 others moves that generally get dismissed by players. However, I am a little worryed that beginners might get the wrong impression. So I am going to put a warning/caution label here.
*WARNING* IF YOU ARE A BEGINNER READING THIS I AM WRITING THESE 2 OTHER MOVES TO BE UNBIASE AND THOROUGH OF FIRST MOVES;HOWEVER, THAT DOES NOT MEAN I BELIEVE BEGINNERS SHOULD PLAY THESE MOVES.
So now I am covered if you lose becuase you are a beginner and didn't listen to my warning label its not my fault. These are complex move's.
1.e3 allows 15 squares to be controlled.
This move is dismissed as bad for its passivity. The fact it does not control more space is 1 of its major critisim.
- Space is considered an advantage so this move does not fight for that advantage which is why it receieves alot of backlash.
- Another reason why this move gets dismissed alot is the fact that white is not forced to play this move. White can play another move with same amount of piece play which is the move 1.e4 while fighting for more space. In a sense the agruement they use here is why play this move when no one is stopping you from playing 1.e4 the better move in comparision lol.
However, through all of its dismissal a student of activity can not question its overwhelming piece activity being generated with this move. This move is in fact the second most powerful square controlling move in the game behind 1.e4. It even surpasses 1.d4.
The reason why I kind of shy away from telling beginners about this move is becuase of its tranposition quality's are very great. This line can tranform into 1.c4 lines, 1.d4 lines, hyper modern lines, 1.f4 lines which beginners may not be able to play by pure lack of knowledge. Also if the position demands a beginner to tranpose into another line to generate more activity they may not know they are suppose to or not. Which can really hurt them. So its complex move.
However, I'm showing it to be "THOROUGH"
Next move.
1.d3 allows 13 squares to be controlled.
This move has more activity than 1.c4 and the knight moves as comparision from before. However, again it is dismissed for being passive and not staking claim to more space which is considered an advantage in chess.
As you can see from the square controll it is generating some square control.
Again the tranposition quality's are very great. This line can tranform into 1.c4 lines, hyper modern lines, 1.e4 lines
Piece activity its a beautiful thing and that is all.
Hope that helps and gives some further insight. Have a nice day.

X_PLAYER_J_X: Don't perceive me as rude, but your argument is invalid. A move cannot be weighed on how good it is by just calculating the number of squares it controls. By your examples, Nc3 is just as good as Nf3, and f4 is just as good as c4. This is far from the truth.
So, your comments are misleading for the chess community. Please come up with a better explanation. Thank you!
(Now please don't block me. Just lol)

X_PLAYER_J_X: Don't perceive me as rude, but your argument is invalid. A move cannot be weighed on how good it is by just calculating the number of squares it controls. By your examples, Nc3 is just as good as Nf3, and f4 is just as good as c4. This is far from the truth.
So, your comments are misleading for the chess community. Please come up with a better explanation. Thank you!
(Now please don't block me. Just lol)
Yes it can be. Nc3 and Nf3 control the same amount of squares. 1.Nf3 is slightly favored becuase it helps the white king to castle faster.
The move 1.f4 is no were near as good as the move 1.c4. from the initial starting position. With the stand point of activity.
1.f4 allows 11 squares to be controlled.
1.c4 allows 12 squares to be controlled.
As you can see 1.c4 is better than 1.f4 in the form of activating activity. Controlling squares.
Is it best to move the Knight on the opening move, or a pawn on the opening move?
Does it depend if your playing black or white???