Downtube shifters will really get you in touch with your bike--more than with many other shifting systems, I believe that I can actually FEEL the rear derailleur moving in my hand when I use them.
Needless to say, I ride a lot of old road bikes.
You will quickly gain the control to shift either from various hand positions on your road bars. Assuming the bike fits you, you should ultimately be able to use the shifters from nearly any position. You will, of course, use the right hand to shift the rear derailleur and the left hand to shift the front.
I will talk mostly about shifting the rear derailleur, as it will require the most control and finesse.
Note also that these shifters each have two main parts that are pertinent here, the LEVER and the BARREL. The lever is pretty obvious--it's the usually-flattened handle that sticks out. The barrel is the round bit which is attached to the frame. The cable wraps around this round bit--as you turn the shifter, more or less cable is wrapped which is how the shifter moves the cable and thus the derailleur.
Ordinarily, when I reach down and the bike is in high gear--with the shift lever all the way forward and parallel with the downtube--I will grasp the lever by wrapping the thumb of my right hand over the end of the shift lever. I press the side of my pointer finger against it and slowly pull back. This motion has less control over precise gearing and is best suited for quickly downshifting before climbing.
If I am already in low gear and the shifter is out away from the downtube, I usually actually grasp the entire lever with my right hand. In this mode, the tip of my thumb is on the barrel and my fingers are all wrapped around the shift lever. Motion in either direction from this position can be very precise. As I said, you can feel the derailleur move if you're cables are new and well-greased.
It's a good setup for shifters, in my opinion. I can't say that I prefer downtube to integrated, but as far as simplicity is concerned... nothing beats it.
Except, perhaps, Sheldon Brown reaching down and putting his forefinger on the chain to move from upper to lower chainrings.
If you get a chance at the library, check out Understanding, Maintaining, and Riding the Ten-Speed Bicycle by Denise De la Rosa and Michael Kolin (Rodale Press, 1979). This book describes shifting and using downtube shifters rather well, right down to the groovy 70's costume. De la Rosa describes another method of hanging the thumb (I think) off of the top tube while reaching with the same hand's fingers down to the shift lever. This only would work on a frame smaller than mine, but it's another piece of cycling history.
I hope this helps. It certainly took my mind off my work for a few minutes!