Yes, that is normal.
Normally only semi-autos have a trigger that is initially double action and after it is fired and cycles, every subsequent shot is single action.
There were some exotic revolver designs like the Mateba, Landstad, and Webley-Fosbery, but they're not your common every-day revolver.
Most revolvers have either single-action only or double / single action. With a single-action revolver, the hammer must be cocked manually, this also rotates the cylinder. The trigger then simply drops the hammer. With a double action / single action revolver, the trigger will turn the cylinder and pull back the hammer if its not already cocked and will drop the hammer in one stroke of the trigger. The double action / single action revolver also allows the hammer to be cocked manually, in which case the trigger simply drops the hammer.