Subdividing polygons

The key for most tesselations is performing them on EDGES. So i suggest you incorporate edge array into your model class as well, in addition to vertices and faces.
So when subdividing a triangle, you can just split one of the edges to half, thus producing one new vertex and three new edges ( two halves of the original, and one from the opposite corner of triangle) and discarding the old one you split.
For fastest operations its best to keep two-way pointers from each data unit ( vertex, edge and face )
In consistent model ( no disjoint vertices ) you will always have:
Polygon has 3..n *vertices, and 3..n *edges
Vertex has 1..n *polygons, and 2..n *edges
Edge has exactly 2 vertices, and 1..n polygons

Now for actual triangle splitting you can use some different methods. The one i described above: splitting one edge, creating new vertex in the middle of edge. If the edge is shared by other triangles, those will have to be split too.
Or, you can create new vertex in the middle of triangle, thus forming three new edges.
The nicest method is to split all three edges of triangles, creating three new vertices, three new edges and potentially forcing splitting of quite a lot of adjacent polygons.

Note that array of edges doesnt have to be saved along with model, you can rebuild it from face info when you need it. Edge information is not necessary for solid rendering either, but can be useful when rendering wireframe.