Welcome to the forum, you’ve gotten great advice. And there are a bunch of great resources to help get you started. Where in NY do you fish?
First, to get a general sense of the basic lifecycles and to be able to recognize examples, here’s a link to Jerry Hadden’s site. He’s a fly fishing guide in the Catskills and his site has a ton of good basic information.
Wild Trout, Delaware River Fly Fishing Guide Jerry Hadden
From his home page, first click on Aquatic Insect Identification and from there look click on the “Life Cycle” tab to get a drop down of mayfly, caddis and stonefly life cycles.
Study them to be able to recognize the general appearance of
Mayfly (nymph, emerger, dun, spinner)
Caddis (different types of cased and free living larva, pupa, adult)
Stonefly (nymph, adult)
Another site that has great info is Troutnut.com Fly Fishing for Trout
, with detailed info and pics of different insects. Here’s a good page to start that shows pics of the main categories of trout food: Aquatic Insects of American Trout Streams
Second, get a hatch chart for the area you plan to fish—many can be found online, and if you have a local flyshop near where you fish they‘ll have one. This will give you a big head start on what to expect on the stream, and you can pick up or tie flies based on what you are most likely to run into.
There is also a ton of other info on Jerry’s site including a hatch chart for the Delaware (which also should be good for most Catskill streams) but it may be too much detail at this point—at this point you should concentrate on the major hatches you’re likely to run into not every possibility that includes sporadic/infrequent hatches.
Most good hatch charts will tell you the insect’s common and Latin name, the hook size, patterns that can be used to imitate it, and time of day they typically emerge so you can be on stream to catch it.
You can then look up specific insects you see on your hatch chart on Troutnut to be able to recognize it on stream or google "fishing the __(whatever)___ hatch” to get some tips on tactics for fishing that hatch.
As for #2, you’d probably want to have some flies that would do a couple of different things- cover you for some of the major hatches that will happen now through the summer as well as be good searching flies when no hatches are going on, and to have a mix of flies that will cover you for different situations (a few streamers, fast and slow water dries, and some wets and nymphs) A good local shop if you have one can help you pick out a selection, and many have stream reports that will give up to date info on hatches taking place as well as flow rates and stream conditions.
Tell us what flies you have now, and maybe we can suggest some additions. Assuming you have many of the usual suspects in your box already, this time of year, many of the darker early season mayfly hatches are winding down and giving way to yellow and cream colored ones. Dries like these would be good to add for most streams in NY (and elsewhere in the east):
Sulphur Sparkle Duns 16-18,
Light Cahills 12-14
Cream Variants 10-12
Ausable Wulff size 12 is a good fast water fly for searching riffles and pocket water. It's also a decent imitation for March Browns and Gray Foxes that are winding down now in most areas.
Some rivers with slower silty sections (like the Delaware) get hatches of Drakes. These are big honking mayflies and specific imitations are required for Eastern Green Drake duns and Coffin Flies (the spinner form of Eastern Green Drakes) size 6-10, and for Brown Drakes duns and spinners size 10-12. These Drakes hatch in June and when they occur it's typically at dusk until dark. These hatches can bring up big fish but are often hit or miss. The good news is that fishing an imitation on streams where they occur can often bring vicious strikes even when no naturals are on the water.
Keep asking questions, and let us know where you are-- I'm sure folks can point you in the right direction.