It is said that the only way to practice fighting big fish is to fight big fish. Most of us only get to test the limits of our tackle very occasionally, so if you need an excuse to start chasing carp with a fly rod, this is a good one. You'll soon find out what backing is for.
Carp are every bit as spooky as a bonefish and as discerning as a diamond merchant.
As a quarry, carp are largely underrated by those who haven't given them a shot. Carp are every bit as spooky as a bonefish, and they can be as discerning as a diamond merchant. Drop a weighted fly too close, or lay a leader over their back, and they are gone in a boil.
Carp eat a variety of forage including vegetative matter, worms, grubs, leeches, crayfish, berries, seeds and generally anything else that passes under their nose.
Nymphs are a natural starting place. You want a fly that sinks pretty quickly, since they are primarily bottom feeders. The larger nymphs in your box that you usually pass over when trout fishing are good candidates for carp as they are easier for the fish to see in murky water. If there is a particular seasonal hatch, like Hexagenias, start there. Dragonfly nymphs are great.
You'll recognize actively feeding carp by their head-down, tail-up attitude as they root around. Determine the direction they're moving, and put the fly down ahead of them. Some anglers will use a small strike indicator. Don't retrieve quickly; impart just enough movement to get the fish's attention. Takes are subtle, and detecting them will take time to master. Eliminate slack, and watch the tip of your line for twitches.
Carp are strong fighters. If you use tippet lighter than 2X, or a rod lighter than a 6-weight, you may be under-gunned. A 7-weight is good, and there are many big water carpers who use an 8-weight outfit.
There are lots of great books and websites devoted to carp fishing with fly gear. The icing on the cake is carp are everywhere. No matter where you live, there will be carp water to explore.