“STI” stands for “Shimano Total Integration” and it refers to the fact that the gear shifter and the brakes can be accessed in the same lever.
How it works is that the lever is a vertical one that if you pull the lever towards you, you operate the brake. If you push it in towards the bike, you operate the shifters.
In order to adjust them, you can do several things:
By this time, if you think you need to adjust your shifters, you probably have a good idea of whether it is the front derailleur or the back one. Check your cables: if they are frayed, rusted or corroded, you'll have to get new ones installed. For that, I've uploaded a rather long video on how to replace that (1st vid) and actually clean the shifter mechanism itself.
If it's not the cables, then it'll be the derailleurs. Adjusting the front and the rear are based on the same principles: alignment and cable tension.
If you hear a weird noise (like a metallic rub) then that may be more a case of incorrect alignment of the derailleur. If so, check out the post on that.
To check for proper cable tension, pull on the cables that are on your down tube or top tube. If you can pull your cables off your frame with any sort of ease, you'll need more tension. Adjust this by using your barrel adjusters. On STIs, check where the cable comes out of the shifters on your handlebars. Near the body of the shifter, there will be a piece that can twist. This is the barrel adjuster, which you need to give a turn or two. Run through your shifting again.
This process is via trial and error. You'll know if you've overdone it when it doesn't shift crisply up and down.