Training for liberty starts with a halter and lead rope. The horse is taught cues with the halter and lead rope on, slowly the cue is made lighter and lighter until the rope can be let drag on the ground or removed altogether.
There are a few trainers that recommend teaching starting at liberty with the round pen as the main tool. I've done it both ways and gotten much better results by starting with the lead rope on. Though some things were faster to teach the round pen way, my horse's attitude suffered and he didn't seem as "happy" about it.
I suppose it might be possible to teach for liberty with a regular web halter and cotton lead, but a rope halter is much lighter allowing for much softer signals, and because it is so light the horse hardly notices the difference when the halter is eventually removed. The longer yacht braid ropes that many of the trainers use are also very useful in transferring the tiniest signals possible to your horse.
If you want to get one of these halters & leads I would recommend the ones made by Weaver Leather, they have Stacey Westfall's endorsement and are more economical than the ones sold by other trainers.
I plan to start posting a few videos of my liberty work soon to get opinions(little sacred) I am taking a NH training exam with my horse the first section is on liberty, I will be asking people what they think looks good or not so I can decide what to include in my exam.