Gyrocopters are a lot of fun, and for the most part, very easy to fly. They're flown more like a conventional gear airplane than anything. The land slowly, they're easy to maneuver, and running out of rotor energy is not really a problem as they're always in a constant state of autorotation.
Some gyros are safer than others, and some are easier than others. The only real danger to flying a gyro, perhaps a little more of a problem on some of the cheaper or earlier designs, is "bunting." From the pilot perspective, this is a tendency to go nose down and dive into the ground; clearly an undesirable thing. It's a pilot-induced situation involving reversal of airflow through the rotor disc. It can be brought on by a hard pushover, and it can be unrecoverable.
Gyros that use a vertical stabilizer are less likely to bunt, and gyros that have the centerline of thrust (axis of the crankshaft, essentially) in line with the center of gravity of the gyro tend to be more resistant to bunting, as well. This is the reason that you'll see a lot of gyro designs out there with the pilot sitting in what looks like an unusually high position, with longer spindly legs; the design has been made to put the center of gravity higher, and in line with, or above the center of thrust.
The center of thrust is naturally high because of the need to elevate the engine to give propeller clearance.
Some designs use a horizontal stabilizer, some don't. The gyro design dictates, but designs not using a horizontal stabilizer tend to be ones that are more "expert" in nature, and ones that are far less forgiving.
You'll also see two basic power configurations; pusher, and tractor. More gyros are found as pushers these days, but the original configuration, and generally the most stable, is the tractor arrangement. This also more fully replicates a fixed wing airplane.
A few gyros have an elementary collective control, and with adequate rotor inertia before takeoff, are capable of a "jump start," or "jump takeoff" that requires no forward roll. The Air and Space 18A was one such gyro, but very few gyros can do this. Most use a "pre-rotator" to get an initial spin on the rotor, then use the forward takeoff roll to get the rotor going the rest of the way. Rolls tend to be fairly short, and landing rolls are extremely short. In fact, one can generally land, and come to a stop, and with a little back pressure on the stick, move backward.
Various degrees of enclosure can be had with different gyro designs. Some leave the user out in the breeze, while others entirely encapsulate the user.
Find a reputable trainer and get an introductory ride. Be forewarned; gyro flying is very addictive, and a LOT of fun.