How we did it:
To make Tangrams you really only need two things: felt and scissors. If you wanted a template you can easily find one online and you can use that to cut your shapes. Or you could free-hand cut them.
Start by cutting out your shapes. Tangrams are made with shapes with straight sides, so cut triangles, rectangles, diamonds, triangles, rhombuses, hexagons, and so on. Cut out a nice array of shapes and in different colors.
Then they are ready for kids to use. They can make their own designs on blank paper or you can print tangram puzzles for them to fill in. My 3 year old liked both, especially making animals.
The puzzle aspect is figuring out what shapes will fill the space without leaving any white space behind. Put the pieces in a bag and you can bring them anywhere!