My first roadbike had downtube shifters. And most of the roadbikes I own now have them as well. I seem to like them. All are friction, and one is a 24 speed, so its doable. Here are some things I do, not sure if it will help or not. My only qualifiction for giving advice is that I do use DT shifters a lot.
can you ride no handed? if so, then try for that feeling when reaching to the shifters. Tensing up on the bar to keep it going strait is harder than being able to relax a bit.(don't take your hands off the bar to shift, just try not to be tense and overcorrect!)
You should be able to shift with your hand on the drops, hood, or top. You should get a feel for what is best when climbing, on the flats or a mild descent(I seldom shift on a fast descent, sometimes, but usualy only if I am totaly spun out in the gear I am in and its a long descent. In that case, I just jam the levers into the top gear with little finess. if its a short downhill, I just wait till I slow a bit on the bottom, usualy rolling hills)
Sometimes I put my thumb or a finger on the tube when shifting, sort of like a feeler gauge, moving the lever with the remaining fingers or palm area. I move my whole hand, feeling the sensation where the finger rubs the tube. after a few shifts, having some sort of contact area helps let me know where I am. I mostly do this because I am changing between several bikes and they all have slightly different shifting positions.
On some of my older bikes with five speed freewheels, I overshift slightly, then move the lever back a fraction. Thats because older freewheels dont usualy have special shaped teeth that help the chain move as quickly to the next cog. So sometimes to get the chain to move, you have moved the lever to the point that it wants to rub the chain on the next cog. THis is just automaticaly trimming that can come with lots of practice. On bikes with more modern cassettes, shifting is usualy more precise and snappy.
When riding with people who have integrated shifters, I always hear chains grinding from indexing thats out of adjustment, or because they don't know how to trim with indexing. I kind of take pride in my quiet drivetrain, so friction downtubes are not that bad. When you get used to friction downtubes, and someday get a more modern system, you will have a better understanding of whats going on with indexing.