Engraving something onto a surface using the knife project tool:
First, create the outline of the shape you want to create with vertices in a separate object:
Then, in object mode, first select the shape, then the cup (or other object) behind it second, then go into edit mode:
Now, using the search menu, do a Knife Project, making sure the viewport points directly at the surface:
Now, you can just extrude the vertices on the cup inwards to create the engraving:
One note is that i used the auto-smooth feature under normals to remove any bad shading artifacts.
You'll need to be a little less vague on the second part. What do you mean by "apply the tribal design"?