TUI, the two-stage "trigger-cocking" stroke has pretty much gone the way of the dodo bird. It worked best with old-style Colt revolvers whose trigger pulls stacked, or grew perceptibly heavier toward the end of the pull, and with those guns discontinued for a decade now, the two-stage technique has faded even more. It might have some value in bullseye revolver match shooting, such as the Distinguished Revolver and Harry Reeves events at Camp Perry, but I don't know of any highly skilled revolver shooters using the two-stage technique for speed work or self-defense.
First, it takes longer: there is the take-up stage, and then there is the final squeeze-off with the last few pounds of pressure. Second and perhaps more important, the two-stage pull is extremely dependent on the sense of touch, and that's one of the first things we lose in a stress situation. That's because vasoconstriction occurs, redirecting blood flow away from extremities such as the fingers. The shots tend to go off clumsily and prematurely when trying to two-stage with an impaired sense of touch.
From Jerry Miculek, the undisputed top double action revolver man in the world, on down, the smart money is on the single-stage trigger press technique. The movement is smooth, even, consistent and uninterrupted. This gives more of a surprise trigger break and reduces anticipation and trigger jerking.
Like any other physical skill, crawl before you walk and walk before you run. Start dry fire. Progress to live ammo, first in slow fire, then in cadenced fire. In time, you'll be able to hit the center of the silhouette faster than an amateur could jerk his trigger and miss it. Dry fire is your friend, and no handgun is more amenable to it than the double action center-fire revolver.
Best of luck,