A B757 FLT CREW CLBING OUT OF JFK DEVIATED FROM THE SID TRACK WHEN THE AUTOPLT MADE A SHALLOW TURN.

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BEFORE START; WE BRIEFED OUR TAXI TO AND DEP OFF OF JFK RWY 4L. THE ATC CLRNC INCLUDED AN ASSIGNED HDG AND CLB TO ALT AFTER TKOF. DURING TAXI OUT; ATC CHANGED OUR DEP RWY AND CLRNC. ALSO DURING TAXI OUT ATC GAVE US INSTRUCTIONS TO HOLD SHORT OF A FEW DIFFERENT TXWYS DURING OUR LONG TAXI OUT TO RWY 31L. OUR NEW ATC CLRNC WAS NOW THE KENNEDY 9; BREEZY POINT CLB TO CANARSIE VOR (CRI). AFTER TKOF AND PASSING OVER CRI AT 3000 FT MSL; THE FMC COMMANDED 300 KTS CLB SPD. I IMMEDIATELY SELECTED 250 KTS ON THE SPD CTL. THE CTL SPD WAS MOST LIKELY PROGRAMMED INCORRECTLY AND I DIDN'T CATCH IT DURING THE PREFLT SET-UP. I FEEL THAT THIS UNUSUAL EVENT LED TO THE FMC BECOMING A BIT CONFUSED AND THEN LEADING TO IT MAKING A VERY SLOW TURN TO JOIN THE 223 DEG RADIAL WHILE IT FIGURED OUT WHERE IT WAS. THIS TURN WAS TOO SLOW FOR ATC AND MAY HAVE CAUSED US TO TRACK TOO FAR W AND INFRINGE ON THE ARR/DEP AIRSPACE FOR LGA ARPT. AT ABOUT THIS TIME; I REALIZED TOO THAT THE TURN WAS A BIT SLOW BY THE AUTOPLT AND TOLD THE FO TO TURN IMMEDIATELY TO A HDG OF 220 DEGS. AS HE STARTED TO INCREASE THE TURN RATE; ATC THEN TOLD US TO TURN TO A HDG OF 180 DEGS. AFTER THAT; ATC GAVE US A HDG OF 220 DEGS TO PARALLEL US WITH THE 223 DEG RADIAL OUT OF CRI. NO OTHER ACTIONS WERE TAKEN AND THE REMAINDER OF THE FLT WAS UNEVENTFUL. I FEEL THAT THERE ARE THINGS I LEARNED THIS PARTICULAR MORNING; SOME RELATED TO THE EVENT AND OTHERS NOT. 1) CHK ALL FMC PAGES MORE CLOSELY DURING PREFLT. 2) WHEN ATC CHANGES THE TAXI RTE; DEP RWY; SID AND CLB; OR ANY COMBINATION OF THEM -- PAY PARTICULAR AND CLOSE ATTN TO THOSE CHANGES. ANY VAGUENESS ABOUT THE DEP SHOULD BE CLRED UP BEFORE TKOF AND A VERBAL BRIEFING BY JUST 1 CREW MEMBER ABOUT THOSE CHANGES ISN'T PROBABLY GOOD ENOUGH. JUST STOP THE AIRPLANE AND READ; CHK; VERIFY AND BRIEF THOSE CHANGES. IT'S TOO LATE WHEN YOU'RE IN THE AIR AND THINGS AREN'T CLR. 3) WHEN THE AUTOMATION IS NOT DOING THE CORRECT JOB OR WHAT YOU NEED IT TO DO QUICKLY ENOUGH; TURN IT OFF AND FLY THE AIRPLANE BY HAND. 4) WHEN YOU HAVEN'T BEEN INTO A PARTICULAR ARPT IN A LONG TIME; MORE ATTN MUST BE PAID TO THE SIDS/STARS DURING PREFLT THAT MIGHT BE ASSIGNED DURING TAXI. HAVING SOME KNOWLEDGE OF THE DIFFERENT ARR/DEP PATHS IS BETTER THAN NONE AT ALL.