A pawn move is the square it lands on, and nothing else. e4. d5. h3. There is no letter for a pawn (Appendix C.3), and that missing letter is how you recognise one: a move with no capital in front of it was made by a pawn.
So the first move of a game is e4, not e2-e4 and never Pe4. The two-square opening push gets no special mark either, despite being a special move. If you meet e2-e4 in a textbook you are reading long algebraic, which spells out the departure square as well. It is legal, it is clear, and it is slow; most people drop it inside a month.
A capture keeps the file it came from
Pawns are the exception to the arrival-square-only habit, because a bare square would not say which pawn. With pawns on c4 and e4 both able to take on d5, xd5 is useless. So a pawn capture puts the file the pawn left in front of the x (Appendix C.8): cxd5 or exd5, and the same for gxf3, axb5, hxg6.
En passant works the same way, exd6, using the square your pawn finishes on rather than the square of the pawn it took, and you can add e.p. if you feel like it. Where the x itself goes is the same for pawns as for anything else.
Promotion: the move, then the new piece
8 
g e8=Q is a pawn reaching e8 and turning into a queen. The equals sign is how apps, databases and modern books do it. The laws write it without the sign, e8Q, and older books use brackets, e8(Q). All three say the same thing and all three will be understood, as long as you name the piece: a promotion with no piece named is an unfinished move.
Name it even when it seems obvious, because it is not always a queen. e8=N with check is a genuine resource, and there are endings where a new queen is stalemate and a rook wins quietly. That is why underpromotion has a puzzle collection of its own, and the rule behind it is on what a pawn can promote to.
Three ways beginners get it wrong
Writing Pe4. There is no P in algebraic notation. The empty space in front of the square is the pawn.
Writing ed5 with no x. Strictly the laws allow it, since the x “may be inserted”, but a scoresheet has to be read by an arbiter in a hurry. Put the x in.
Writing the captured pawn’s square after an en passant capture. It is exd6, where your pawn lands, not exd5, where the pawn you took was standing. That one has its own page, because it catches almost everybody once.