From a two-handed firing grip
With a S&W in your strong hand, bring your left thumb around the backstrap to push forward on the thumbpiece to release the cylinder. Simultaneously, your right support hand moves forward on the frame slightly so that its thumb can shove the cylinder open and wrap all of the fingers around the now open frame window. The thumb continues to hold the cylinder open to prevent it from closing and obstructing the reload.
Alternatively, you can shift your grip slightly as the support hand takes over and use your trigger finger to push forward on a S&W thumbpiece, in on a Ruger's or pull back on a Colt's. With a Dan Wesson, the right hand index finger pulls down on the release as the hand wraps around the frame as described.
Now, invert the muzzle up and tap down on the ejector with your palm. This should smartly eject the cases with no hang-ups. Use your strong-hand for the loading task, be it with a loader, or individual rounds. That's the most fumble-prone task, so it benefits from the dexterity of your strong hand.