Everyone makes these three mistakes when fly fishing. Fortunately, they are simple to correct and can improve your chances of not spooking any trout.
1. Rushing
We're so eager to fish the pool that we sprint, ignoring and spooking all the trout along the way. Trout are acutely aware of vibrations and sounds that occur within a river, some better than most. So stumbling on rocks and thundering upstream does not help your odds of catching any.
Take it slow. Slow down your pace and fish up to the hole. Remember you're technically a hunter, so be stealthy.
2. Too Much Line
This applies both to nymphing and dry fly fishing.
Having too much line introduces a whole set of problems. I have to admit, trying to push and see how far you can cast like in "A River Runs Through It" is a good time. On the other hand, when you slap 40 feet of line on the water, trout are going to know.
To avoid this, try to wade closer. Shorten your casting distance so you can give your fly a better presentation.
3. Diving Right In
Before your line gets wet -- take a few minutes and see whats going on around you. Some of the best anglers I've ever fly fished with take nearly 10 minutes to observe their surroundings.
Look above you for any hatches. Take note on how the water moves around obstructions. Look to see how you would enter the water to avoid spooking any fish.