This polyhedron can be made in only 3 colors or in 10 colors.
To make the shape in only 3 colors, use one color for all of your triangles and 2 colors for your squares. As you build your polyhedron, no two squares of the same color will be touching. You will need 8 triangle faces and 18 square faces, 9 of each color.
To see the full-sized template click on this image:
To make this polyhedron in 10 colors, you can again make all of your triangles the same color. You will need to make 8 triangular pieces. But your squares will be made of 9 different colors. You will need to make 2 square faces of every color. Choose one square face as the center of the base of this polyhedron. Attach 4 square faces of different colors to the edges of that central square. Connect those 4 attached squares by adding triangular pieces between them. This completes a bottom bowl of the polyhedron.
The next ring of the polyhedron uses just 8 squares in 4 colors. Use a new color square to attach to to the bottom bowl of the polyhedra. The same color square should be placed directly across from your first square. Continue in that manner until the ring around the bowl is complete with pairs of squares.
The pieces of the top bowl of the polyhedron are placed directly across from and in parallel planes to the pieces in the bottom bowl.