You've probably heard of the expression "turning over a fly". The goal with casting your fly line is for the unfurl completely, or "turn over", straight. If it doesn't do this, the fly plops down into the water in a nest of line, which is less natural looking and can cause you to miss strikes while pulling in the slack line.
Your fly line acts like a bullwhip, graduating (or tapering) to a smaller and smaller diameter until the end. This allows the line to force the fly out and settle on the water naturally. A tapered leader goes from about the thickness of 30lb mono down to 4lb flouro at the fly end (the "tippet"). You'll definitely need a tapered leader. If you want to save a few bucks, cut the last 18 inches off and retie it back on the leader. That's the part you'll use up ALL the TIME due to tying on flies. I tie a loop knot in the leader, and, when it gets too short, I tie on another piece of 4-6lb flouro to replace it instead of replacing the whole tapered leader.