I am interested in upgrading a bike to a TRIPLE Shimano Drivetrain from double.

Want to know if the following 9/10s combination will work ok? Particularly will the Front chainrings shift ok with the 6603 Crankset and 9s chain?

Mix of components would like to use:

RD-6600 GS Long cage Rear derailleur
FD-6603 Front derailleur
FC-6603 Triple crankset & bb
CS-6500 9 speed cassette
Shimano 9s hg93 chain
Shimano 9s 105 STI levers.

Thanks very much for any help/comments.

They should work together, but you need to change the bottom bracket to the one designed for a triple crankset to get the chainline correct. Are you planning to eventually go all 10 speed? If not, why not get 6500 parts instead of 6600?

you need to change the bottom bracket to the one designed for a triple crankset to get the chainline correct
Huh? The 6603 crank includes the axle. Just need the SMFC6600 bearing set, which is double/triple compatible.

Absolutely right! Just a momentary brain fart.