The ST-7700 brifters will work fine with your RD-6500 rear derailleur. The mis-match may come with the front shifting. Do you have a triple crank as your front derailleur implies?? 7700-series STI's came in double OR triple versions so you need to have the corresponding model.
STI's aren't hard to set up. Look at the Shimano tech documents web site for a detailed guide.