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There are two casts that you need to know. One is the Overhead Cast and the other is the Roll Cast. There are many other casts but these two will get you through most of the situations you may encounter fly fishing. To practice these, the best way is to try and do them in an open place with still water, like a pond. You want to find a spot without any trees or tall plants.
Before we start - Right-handed casters with hold the rod in their right hand and pull line off of the reel with their left hand. Grip the rod on the handle with your fingers around the grip and your thumb pointed toward the rod tip on the top of the rod. Right handers generally wind the reel with their left hand. Lefties do the opposite of righties.
Start with some line out on the water straight in front of you. The length of line should be 1-2 times the length of the rod.
Step 1 - The Back Cast
The back cast should be a single, continuous, swift motion. You do it by lifting upward slightly and rotating your arm back. For a novice angler, watch your rod tip as you make your back cast. You should stop your back cast when your rod tip is just past vertical. This is the 2 o'clock position. When you’re in this position watch for your line to straighten all the way out behind you. When it is straight you'll feel the total weight of the line pulling the rod. This is what is called the "load". Understanding and feeling this “load” will come with practice. Notice that where you point the rod tip where you stop controls the direction the line travels.
Step 2 - The Forward Cast
When you see the line straight behind you bring the rod forward quickly and stop abruptly at the 10 o'clock position. A great way to master your forward cast motion is to imagine your rod is a hammer and you are using it to hit a nail on a wall in front of you. Where your thumb points is where the line will go.
This is arguably the most useful and underrated cast in fly fishing- master this cast.
Here is a video of the Roll Cast:
Here are written instructions with pictures:
Step 1 - With the line in front of you on the water, slowly lift your rod tip up towards the sky while slightly tipping the rod behind you. As you do this motion the fly line drooping from the rod tip will begin to make a bow or loop heading behind you.
Step 2 - When the drooping fly line crosses behind the vertical rod make the same forward stroke you would during the overhead cast. The line will follow the loop and lay out in a straight presentation. It is a very simple cast but can be used in a lot of situations.
Source: Fix.com Blog
Take your rod to some still water and give these a try. Once you feel like you get it right as much as you do it wrong, go find the streams in your area. Don't get discouraged. Even very experienced fly fishers occasionally end up with a tangled ball of fly line and tippet.