I've also been a little frustrated with the shutter lag on the S4. I was standing next to an iPhone user at a concert recently and was impressed how quick the iPhone was to snap a photo, almost instantaneously after touching the screen, compared to my S4.
If you're having trouble timing the photo just right, you might consider turning on Burst Shot in the main Camera settings, which takes a burst of shots as long as you hold down the button. This is good for fast-moving kids and sports when you're hoping to get one exact moment. [Note that those photos by default get saved to internal memory, not your SD card.]
While you're in the Settings menu turning on Burst Shot, you may also want to make the volume button your camera shutter button. That way you aren't awkwardly trying to hold the phone perfectly still while tapping the screen to take a photo. You can change this by going into Camera, Settings, gear, Volume Key = the camera key.
Another related option is "Best Shot" Mode, which takes 8 snaps in rapid succession and asks you to pick one to keep.