Your present rear derailler will work fine. You'll need a 10-speed brifter, which makes that vital 10th click that you 9-speed doesn't, and a 10-speed chain, which has a narrower OD than the 9-speed. This would be a good time to upgrade to a better cable, since it all has to come apart anyway.
Once the new bits are in place, tune that rear derailler, and you're good to go.
If your crankset is a double you need new 10-speed shifters, 10-speed cassette, 10-speed chain and bar tape. If you have a triple crankset you may or may not need a new 10-speed crankset. The 9-speed derailleurs will work with a 10-speed drivetrain.
BTW, 10-speed shifters click 9 times and 9-speed shifters click 8 times because the first position is with all the cable out.