Keeping score with algebraic notation is easy! Most 7-year olds can do it, and it is required (if you know how) in US Chess Federation (USCF) tournaments. There are many chess equipment sellers who sell scorebooks for ~$3 apiece to keep track of ~ 50 of your games (or just use a blank page).
Advantages of scorekeeping:
1) Replay your games to show better players and learn from your mistakes,
2) Re-set positions after illegal move sequences,
3) Prove who is to move and what the position is, and
4) Claim three-fold repetition and many other rules that require scoresheet proof.
To record a game ("keep score"):