In my Occhi Beginner’s Course Tatting Basics 1 - Flipping Knots and Your First Ring , the main focus is on forming the knots: the “jumping” over and under knots are quite a lot to take in for a beginner. At first, the most important thing is getting the tension of the thread in your holding hand right. Exactly how you fiddle the shuttle thread through the loops doesn’t really matter in the beginning. Here, I’ll show you how to hold a shuttle most effectively and what else to pay attention to.
There are certainly many ways to handle a shuttle, and each has its own advantages and disadvantages. Here, I’ll explain one common method* using a ring as an example:
The shuttle never flips or turns. Instead, it glides back and forth in a straight line, just like a shuttle in weaving. Whenever possible, it stays in contact with the ring thread, using its flat side (not the open side). Dabei hält es, wann immer möglich Kontakt zum Ringfaden und zwar mit der flachen Seite (nicht mit der offenen Seite).

When making the under stitch the shuttle is not simply passed under the ring thread; it also goes under the shuttle thread first. I find this easiest when I hold the shuttle with the thumb and index finger of my working hand around its outer half, with the thread coming out of the shuttle toward the back.

The middle and ring fingers of the working hand reach over the shuttle thread and pull it back and slightly upward, forming a loop.

On its way back, the shuttle passes over the ring thread but under the shuttle thread.
When making the "over-stitch I hold the shuttle thread out to the side with the little finger of my working hand and guide the shuttle over both the shuttle thread and the ring thread. You can also use the ring finger for this. On the way back, the shuttle passes under the ring thread and over the shuttle thread.

And that’s actually all there is to it. Sometimes, a short explanation is all it takes to make things click.
*Among others, this method is described in the book „Klöppeln und Occhi“ by Jutta Lammèr, published by Otto Maier Verlag Ravensburg in 1976.