As someone who's been doing it for decades, I've probably forgotten what it is that I'm doing. The one thing that comes to mind is that you need to reduce the pressure on the pedals when shifting. i.e. only pedal at half-power or so when making a shift. This applies to indexed shifting too, but isn't as obvious.

Back in the day, when Schwinn stores were trying to educate customers about this fancy new 10 speed stuff, they actually had a tiny bike frame on the countertop, with a whole derailleur system installed. The customer could turn the cranks, move the shift levers and watch the chain move between cogs, etc. This reduced the fear factor, got the customer over the first hurdle, and made it easier to then get the customer and let them practice in the parking lot.

It might be good for you to put the bike in the workstand, turn the cranks, and get a feel for how far the levers have to be moved in order to shift from gear to gear. You'll get a feel for the noises involved, how quiet it is when the chain is in the right place, etc.

I don't think there are any real secrets, but there is surely some sort of learning curve. It may take a little while, but you'll get there.

Steve in Peoria