Part 1:
Most of a chessplayer’s skill lies in his ability to see a tactical or strategic pattern, store it in memory, and then when recognizing the same pattern in the future, to be able to recall what type of move worked before and put it to use again.
If pattern recognition is the most critical element of chess skill, then learning as many patterns as possible is the most efficient use of one’s limited chess study time. There are dozens of anthologies of tactical puzzles out there—you got off to a great start with the Fischer book because it stresses checkmating patterns. After Fischer Teaches Chess, I’d suggest Chandler’s oddly-titled How to Beat Your Dad at Chess, which categorizes several different checkmating patterns. If you can read the old-fashioned descriptive notation — chessplayers benefit hugely from being multilingual — then Renaud and Kahn’s The Art of the Checkmate is a classic.
Part 2:
The best way to practice chess is not playing chess. The best way to practice chess is to play through an annotated master game, covering the master’s moves, and guessing them as you go along. Make your move on your board, then uncover his move. If you’re right, yay. If you’re wrong, take your move back, make the move the master actually played, and think about why they differ.
The greatest chess teacher ever, Cecil Purdy, first described this practice in the late ‘20s. His original piece is reproduced here: http://www.tuirgin.com/2009/04/07/playing-with-ghosts/#more-239
Part 3:
I know enough basics to hold my own against average players
The average player in your neighborhood or your family or your workplace is much worse than the average tournament player, who is terrible.
develop your pieces toward the center
No. Develop your pieces with threats.
castle
No. If there is not a threatening developing to be made, then castle.
try to see at least 3 moves ahead
Er, not exactly. If the pieces are in contact, then your job is to see far enough ahead to a static position, then evaluate. If the pieces aren’t in contact — that is, no threats present — then there’s not really a need to look ahead.
look for and exploit pins and forks
This is part of the next.
make aggressive moves your opponent must respond to
It’s really important to get this: To play chess passably well, not only must you see all the threatening moves — all of your threats, and all of his — you must also recognize the unreality of their unreal threats. Yes, you’re looking for threatening moves “he must respond to”, you have to understand that what he’s looking for in reply is some way to ignore you. When you make a move, the first thing your opponent says is “What’s he threatening?” and then “Is that a real threat? What if I ignore it?”. So that’s also what you have to say every time it’s your move — weak player see a threat, and dance like the comic relief character in a Western dances for the bad guy with a six-shooter. Strong players take a stand like Gary Cooper, and don’t budge.
win the exchange
Sort of. Winning material is usually groovy, but since most exchanges are materially equal, it’s vital that you understand this: The side who wins a materially-equal trade is the side whose pieces develop as a result of the swap.
I learned long ago that a lot of players don’t even bother learning the basics.
Most players don’t know ‘em. You think you know ‘em, but you don’t.
On rated sites I average between 1500–1600, yet I can’t move beyond that range. And what does that rating mean anyway?
Doesn’t mean jack. Every chess organization and every web presence that enables one to play chess with others rates games differently, so one’s rating will vary by hundreds of points from place to place. Online ratings, especially, are bogus — they’re deflated because of cheaters, and inflated because players are so often distracted at home, or by a entire web of distraction that is one click away.
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