If you are practicing catch and release fly fishing, the key is to help the trout off the fly as gently as possible. I like to hold the line taught with my left hand while dipping my right hand in the water. Now wet, I then take my right hand and gently but firmly support the trout by the belly. I then slide my left hand down the line and grasp the fly. An easy turn to the right or left is all it usually takes to remove the fly from the fishes mouth. Then simply loosen your right hand and let the fish swim away. A great moment for you with the least amount of trauma for the fish.
If you do want to take a picture of your catch, support the head and place your thumb over the nose with one hand. Use your other hand to support its belly. Gently hold the fish over the water (in case it wiggles away) and smile. Never keep a fish out of the water for too long. Remember, it’s a whole new world for the fish and is a bit like a human being dunked in the water without first taking a breath.