I don't know about the current lot. My bike is at least a dozen years old. But it has STI and it is wonderful. The shifting works just fine and of course this means it works fine for several years.
The only downside is weight. They do weigh more than downtube shifters.
I'm sure the bike shop showed you how they work. Way back when I got mine there was one 'problem' but it really only applied to racers. A tap on the small lever moves one chain smaller either front or back. This means it is harder to jump 3 or 4 gears in the back to jump away. Not a concern for non racers. If it is I think the Campi system allowed you to accomplish this in one shift. (I find I can shift pretty quickly with repetitive taps, more than fast enough for anything besides serious racing).
The STI brake/shift levers work very well indeed, and provide a larger surface for your hands when you're "on the hoods". You can also shift while standing. By and large, they are trouble-free and reliable.
but don't forget the cost. even the older model 8 speed stis are about 150.00 used on ebay. I lucked out and got a set of 105's that were almost new at 50.00 buy it now but that was pure luck and I have been hunting for several months.
I love my STI shifters, however one of the downsides I've experienced is that they're not as easy to self-service compared to the old downtube shifters. I suppose increased functionality usually means increased complexity. Also, if you have cable stretch, it used to be so easy to just put the downtube shifters in friction mode and continue to go. With STI shifters, I have a hard time making micro-adjustments on the fly.
Keep in mind that what makes indexed shifting possible is advances in the design of derailleurs, chains, and particularly cassettes so that they shift a lot better than they used to. Old bikes didn't have index shifting because you had to finesse the friction shifter, overshifting a little and then correcting, to get them to shift.
If your current bike has friction shifters, you will probably have to upgrade your entire drivetrain to use STI shifters -- chain, derailleur, cassette. You'll probably also need a new rear hub to take a modern cassette.
Sprocket man is right on. The new stuff is great when it works, the problem is they go out of adjustment easily and some folks don't know how to fix them. I have both and way prefer the friction over STI due to it's simplicity and rarely going out of adjustment.
Suntour had the absolutely best friction shifters ever made and I still use the Superbe system and know for a fact that this stuff shifts as fast as the STI stuff. The only thing that makes mine a bit slower the STI is that I have to reach for the downtube shifter, but once mechanically engaged it's just as fast and just as accurate...accuracy of course come with over 30 years of experience using friction. Your barend activated shifters though would be just as fast as STI.
Also the older friction stuff used wider chains and gears then todays crap...err sorry-stuff, and the older chains and gears last at least 3 times longer then the new. I've heard from plenty of forum members here and other places that said they replace their chains and sometimes sprockets every 3,000 to 5,000 miles...THAT'S NUTS!!! My chains average 15,000 miles and my last Suntour Winner freewheel lasted 45,000 miles!
No wonder most touring bikes use the simplier friction stuff.
And if you think your new stuff is so reliable wait till the electric shifters come out. Very few will be able to work on that stuff and if your batteries go dead or they freeze or a motor fries on a ride you can't shift at all! Oh boy I can't wait for those and spend gobs of money for the components and repairs and batteries! All this technology spin reminds of the direction the auto industry has gone; this is all being done so that LBS can make more money on parts and labor just as it was done in the auto industry.
By the way you can pick up all sorts of friction gear on E-Bay and some on-line stores for less then newer STI. Superbe Cyclone and VG series is an excellent mid line value, Cyclone was lighter of the two but both outshifted any other companies top line stuff.