Here are the stepbystep instructions for building a Zome model of a Cosmogram, which is one of the ways to nest the five Platonic Solids. The nesting order is, from inner to outer, icosahedron, octahedron, tetrahedron, hexahedron, and dodecahedron.
This configuration was name by mathematician John Conway (of Game of Life fame) in honour of Kepler. The Zome product Kepler's Obsession also builds this model, but in more colors.
For a regular Zome version, you need the following parts: 38 nodes, 30 B1, 30 B2, 12 B3, 12 HG1, 12 HG2, 12 HG3 struts.
Step 1: Build a B1 icosahedron with 30 B1 struts and 12 nodes.  
Step 2: Build a roof over two adjacent faces using 2 HG1, 2HG2 struts and 1 node. Note that this roof is a tight fit. Make sure all the connections are tight!
Add another roof beside the first roof, using 2 HG1, 2HG2 struts and 1 node. Note that short struts of the new roof connect in a straight line to the long struts of the old (and vice versa). 

Step 3: Build a HG1+HG2 octahedron around the icosahedron by continuing to building 4 more step 2 roofs, using 8 HG1, 8 HG2, and 4 nodes. The 6 roofs form three opposite sets, each perpendicular to the others. This step and step 2 are the most difficult part of the construction.  
Step 4: Build a 2HG3 tetrahedron around the octahedron, using 12 HG3 struts and 4 nodes.  
Step 5: Build a B3 cube around the tetrahedron with 12 B3 struts and 4 nodes.  
Step 6: Build a B2 roof over one face of the cube with 5 B2 struts and 2 nodes.  
Step 7: Build a B2 dodecahedron around the cube by building a step 6 roof for the five remaining faces of the cube, using 25 B2 struts and 10 nodes.
All done! 