Medium is what the plants roots will be growing in.
Coco coir is a good option to start with. It can be purchased in compressed blocks that need water or loose and ready to use. Mixed with perlite to improve drainage and aeration in a fabric pot works well with a saucer underneath. It is used in a drain to waste systems. Nutrients can be mixed and poured into the coco coir perlite mix with a watering can. Or a reservoir can be set up with a pump to drip over medium. Cannot be used in a recirculating system.
Clay pebbles are used in hydroponic systems with water. There are many different types of set ups and they require transfer or air pump or a combo of the both to drip nutrient solution onto the clay pebbles.