The technical compatibility: Shimano STI shifters work only for the number of speeds for which they were designed (although there are some tricks to use 8-speed with 7, and 9-speed with 8, and the like... but there's no way to make an X-speed STI shifter compatible with (X+1)-speeds.

Shimano rear deraillers that were designed for indexing all have the same cable-pull ratio and so a 105 rear derailler from 1987, designed for indexing on a six-speed freewheel, and be mated with a current 10-speed STI lever on a 10-speed cassette. (This excludes pre-9-speed Dura Ace, which had rear deraillers with a different cable-pull ratio. Incidentally, 9-speed shifters will work on 8-speed cassette if you use the 8-speed Dura-Ace rear derailler, which had the weird cable-pull ratio.)

So, if you want 9-speed - and I agree with others here that it's generally not worth it for you - you'll have to get STI levers designed for 9-speed. You can keep your deraillers. You can also keep your chainrings - the 3/32" width of all nearly all chainrings made in the last ~40 years will hold a 10-speed chain, even. The difference in chainrings designed for 9- or 10-speed drivetrains is not the width of the teeth, but the spacing between the chainrings. Even so, people use cranksets designed for 6-8 speed drivetrains, and use them with 9- and 10-speed drivetrains all the time without problem.