This is a great question. One way to start is to brows and site like the Umpqua link, or the online catalog of places like Orvis, just to get an idea of what different flies look like. The names can be very confusing but a good place to start would be to learn the difference between different broad categories of flies and some popular patterns that give a good representation of different styles:
Dry Flies, traditionally tied with a stiff hackle collar wound around the hook like an airplane propeller (examples Hendrickson, Adams, Blue Winged Olive) but now also tied with hackle wound around the wing like a helicopter (Parachute Adams, Parachute Blue Winged Olive Etc), without hackle using splayed deer hair wings (Blue Winged Olive Comparadun, Sulphur Sparkledun etc.) or Cul De Canard (CDC Blue Winged Olive) for flotation. There are also several styles of flies like heavily hackled Wulff’s, Humpies etc that are good attractor patterns in fast water. Typically mayfly patterns are tied with upright wings, or “spent wings” for spinner patterns (Rusty Spinner) that imitate the adult stage of the mayfly. Caddis and stonefly imitations are tied with “down wings” (Elk Hair Caddis, Fluttering Caddis, Stimulator). Once you have a sense of the different styles of flies you can mix different styles of flies in the size and color scheme to match the naturals using th pics here on troutnut.
Wet flies- traditionally have soft (rather than stiff) hackle that flows back to the hook point. Typical patterns would be “soft hackles” (Partridge and Orange), Quill Gordon Wet) and more modern patterns like Emergent Sparkle Pupae.
Nymphs- pretty obvious, but take a look at stonefly nymphs (Kaufmann’s Stone) vs mayfly nymphs (Pheasant Tail Nymph, Gold Ribbed Hare’s Ear) vs Caddis larvae (Green Rock Worm, Peeking Caddis) vs Midge larvae and pupae (Zebra Midge)
Emergers- often look like a nymph with a wing of some sort and are typically designed to float in the film- (CDC emrger)
Streamers- can be made with feather wings (Black Ghost), bucktail (Black Nosed Dace. Mickey Finn), and spun deer hair heads (Muddler Minnow), rabbit strips (Zonker) and marabou (Wooly Bugger, marabou muddler)
Hope this helps a bit.