First, clay must be gathered from a clay floor tile. You need a kiln or magma kiln, a dwarf with the pottery labour, and you need to designate one or more clay floor tiles with as a clay gathering zone. Once all these exist, you can order clay collection through the workshop or manager interface which will produce clay boulders that will be stored in stock stockpiles by default. A single clay floor tile will provide an indefinite supply of clay. Once you have some clay boulders, they can be turned into crafts, statues, jugs, large pots, used in construction of workshops and walls, etc., again by creating jobs from the workshop or manager interface.
I asked in which cases do you prefer to use clay instead of wood and stones.
I am a dumb sometimes.
I personally like to maximize value where possible, so I only make from (fire) clay the things which they cause to be more expensive than other materials: bricks for constructed walls where value is not a chief concern (i.e., general construction in the caverns or mines as opposed to part of an important room), hives, jugs, pots.
I arrive at this conclusion by basically following a flowchart for generating excess utility and wealth once the current needs of my fortress are met.
Weapons grade metals; i.e., adamantine, silver, and steel, are made into extra armour and weapons as befit their type; i.e., adamantine battle axes and mail shirts, silver maces and warhammers, and steel plate; in the event that I have twice as much arms and armour as needed by my military, then I will start using excess silver and steel to create large, serrated disks and spiked balls. Weapons-grade bronze becomes trap components; copper in excess of bronze-making becomes crossbows and bolts. I never use adamantine for anything but battle axes, flexible armour, and if I've filled all of those requirements then I will make adamantine clothes exclusively for military use.
Non-weapons grade metals are used in excess to make flasks and goblets. Aluminum and platinum are special cases: I reserve aluminum for making furniture in important rooms; i.e., armour stands and weapon racks in the barracks, that sort of thing; and, I reserve a small stockpile of platinum near a forge exclusively in the hope of getting platinum maces or warhammers from moods. Everything else is just training fodder.
Stone is used primarily for mechanisms. Non-important furniture is made from stone, like most fortresses.
Wood is used for necessary beds, minecarts, and wheelbarrows; excess is turned into spiked balls.
Glass becomes the primary furniture material once I get a magma furnace, and until legendary skill my glass makers are always churning out large, serrated disks and spiked balls.
And finally we reach clay. I never go through the effort of making anything but green glass, so ceramics are more valuable for the aforementioned bricks, hives, jugs, and if I have fire clay then also pots.