If you are down a pawn, see if you can get a position repeated 3x or get a perpetual check. You can always play another game. Chess is interesting when the agressor has to watch out for that sort of thing.
Another tactic when you don't have many pieces left is to move all your pawns til they are blocked, then sacrifice your pieces so that your opponent stalemates your king or loses material declining. Nothing funnier than a desperado queen following a king around a point blank giving checks with no backup.
It is not about just checkmate. It is about you being able to force an outcome. Enjoy.
I am rather new to chess and am not clear on when I should be playing for a draw. I generally resign when I have what I believe to be an unwinnable game. What is the general rule in terms of plodding on for a draw? Maybe the idea of playing for a tie just doesn't excite me. Any advice would be appreciated.