Dig up dirt spots with a hoe
Break clay nodes in the Mines or Skull Cavern
Till artifact spots on your farm, in town, or other areas
Open geodes
Use the Hoe on the beach, forest, or mountain dirt patches
Farm clay from worms/forage spots when they yield clay
Use the Statue of Endless Fortune only if it gives clay through gifts or rewards
Recycle trash with a Recycling Machine for a chance at clay
Dig in the Desert or Ginger Island dirt patches with a hoe