I'm not the most scientific angler out there (although I have used their lines from time to time...), nor would I ever try to lay claim to being the world's greatest fisherman. I can say that I doubt if anyone has more fun fishing than I do (as much fun yes, more, never).
My approach is much the same as others have stated.
I first try to determine what the fish are doing. Are they hitting at the surface, in the film or are they holding to the bottom. I then try to make an educated (or just wild) guess as to what they may be eating. What size of bug they're looking at and as best as I can gauge the color.
If they are eating something off the surface, and it's dark, I'll most likely tie on an adams of about the right size and give it a shot. If they are eating bugs that look light, I'll tie on a light color fly. etc.
For dry flies, basically, I have light color and dark color flies in a variety of likely sizes. I also carry a wide range of flies, but I tend to stick to generic light and dark dry flies of an approximate size to what I think they're eating.
For nymphs, I'll try a hare's ear or a prince or a pheasant tail (if one doesn't work I'll try the other) until I find out what they are keying in on. terrestrials sometimes (beatles, ants, hoppers, green weenie, etc.).
On tail waters, I start small and go smaller until I either can't see the fly, or start getting strikes.
I approach bass and bluegill the same way.
Nothing scientific about what I do, it's try different stuff until I find something that works. I usually catch fish. Some days it's 3 or 4 some days it's 50 with most days being somewhere between and on some days I get skunked.
I have equal fun on all of those days because I just love being out there fishing. It's not about catching fish, for me, it's about fishing.