Now hang gliding is a sport that I have dabbled in some years ago. The main thing in common is that the pilot does hang underneath the wing of both kinds of aircraft!
However, this sport is very different from paragliding in some respects.
If you are learning to fly, make sure to check with your instructor before making a major purchase...
For a start, hang gliders are rigid-winged craft that in most cases have material stretched taut over an assembly of spars. The pilot hangs in a harness from near the center, and holds onto a horizontal bar to control the craft in the air. When the pilot shifts his or her weight, the craft can be made to climb or dive and turn from side to side.
These gliders are transported on the roof racks of a car. Many motorists have seen the long narrow bag on another car and wondered what was inside!
By contrast, paragliders look and fly something like sports parachutes. The pilot hangs in a harness attached to the lines, and controls the craft via 2 'brakes', one on each side. Pulling one brake line at a time will turn the paraglider, while pulling both at once slows down the forward speed and descent rate of the paraglider just prior to landing. The basic flying skills are easier than those for flying a hang glider.
Transporting a paraglider is just lugging a large back-pack around! They can be put in the boot (trunk) of a car or even taken on board a plane flight.
As far as being able to fly free like a bird, the sensation is similar in both craft! Hang glider pilots will claim that they are the most bird-like of all. I kind of agree, having flown both kinds of gliders myself. Each form of soaring has its own special attraction though.
Both types of aircraft are foot-launched too. No wheels. Just lean into the breeze, take a few steps and part company with the ground. Quite often though, the weather isn't quite right and the local sites aren't flyable.
Some pilots cope with this situation by using some good flying simulation software!