Much in the way of fly fishing wisdom has been offered in this thread, and the only place I'll make an argument is with regard to spinners inherently catching more trout than flies. That said, I have a few suggestions that may help:
--Learn about drag. With spinner fishing, your goal is drag, which is when your lure is being pulled against the current (which turns the spinner blade). For most fly fishing, most of the time, you want a dead drift, on the bottom. After I've been spinner fishing for a trip or two, I find it hard to get back to the dead drift of fly fishing. With spinner fishing, it's all about dragging the spinner through where you think the trout is. With most fly fishing, it is all about placing the fly at a spot upstream from where you think the fish is and getting it to sink to the fish's level, even as the fly looks like any other bug or piece of debris floating down the creek.
--The back cast so prominently featured in "A River Runs Through It" looks really cool, but in these parts, it is useful about a quarter to a third of the time. Master the roll cast. Learn about the bow and arrow cast (there's a cool Joe Humphreys video on youtube showing the roll cast). Most importantly: Master the roll cast (repetition intended!).
--If you are catching fish with spinners but not with flies on the same day on the same creek, are you getting hits on the fly but not setting the hook, or getting no hits? Either way you need to check your presentation and line management. If you are getting no hits, likely your line is hitting the water hard and putting fish down. Strictly speaking, the less line on the water, the better, because it casts a shadow and puts fish down. Best to have it drift over shallows and unproductive water. If you are getting hits but not hooking, likely you are not collecting your line as the fly drifts (that is to say, not keeping your line tight). This is like when you make a long cast with that Panther Martin on a windy day and the trout grabs the spinner on the fall, but you have too much line out to set the hook.
--Finally, it's a movie, not a fly casting lesson. Fly fishing looks cool, but it looks extra cool in that movie. For example, shadow casting, where the fly suspends in the air for a split second and the trout leaps out of the water to grab it, is essentially the same thing as using the Force from Star Wars to open a door or knock a battle droid out of action. Likewise, long casts are impressive to watch, but line management and hooking ability on long casts is extremely difficult.
The old school rule for starting out is to have a leader about the length of your rod, plus tippet (seven- to nine-foot tapered leader plus two feet of tippet for nine to 12 feet of leader). For most fishing, you will want to have a rod length to maybe a rod and a half length of line out. Rod length, plus line out, plus leader length means you are about 20 to 25 feet from the fish you are after. That means you needs to think about your approach to the hole, as you would with a spinner, but your approach needs to be inherently different from when you are tossing a spinner because of the whole drag vs. no drag thing.
From here, there are many more chapters, refining leader making, rod length and action choices, micro-drag, dry fly fishing, true wet fly fishing and streamer fishing, Euro/tight-line drifting, not to mention a million other variables. This is why trout fishing is so addicting. The more you learn, the more questions you have.
