Notation

Of course it's silly to write "p" for pawn moves. When a pawn moves from, say, e2 to e4 at the first move of a game, we simply write either "1.e4" (short notation) or "1.e2-e4" (full notation). When a white pawn, staying on, say, f4, takes a black pawn on e5, we simply write either "f4xe5" (full notation) or "fe" (short notation).

For pawn captures I also write simply the two letters )) I use mostly shorthand notation; but when I write comments with diagrams to some game, I use the longhand notation.

FIDE rules 8.1a:

"In the course of play each player is required to record his own moves and those of his opponent in the correct manner, move after move, as clearly and legibly as possible, in the algebraic notation (Appendix C), on the ‘scoresheet’ prescribed for the competition."

Should you insert x and +?

Appendix C.9

So, basically, it all depends on how clear your handwritting usually is, and on how badly you want to make sure you can set a valid claim based on a position proved by your scoresheet...

As for me, I write so: 1.e4 - d5 2.exd5 - Dxd5 etc. Good, old and classic! (including + for check, and = for draw proposals)

BobbyTalparov

You tell that to the FIDE, ok? it's all copied from their official pages.

It seems FIDE and USCF - for whom I've been a Tournament Director, require a record in where it's clear how each piece moved for every move.  In USCF play, "check" does NOT need to be announced when it occurs (although it usually is for psychological effect) since announcing it or not, the K is in check, his owner must recognize it and must move out of it.  Similarly, listing check or mate on a scoresheet does not alter the record so should be ok.

All said already loud and clear for the written part, quoting FIDE rules 8.1a and Annex 13.C.