When I give the faces in the inside a differerent material than the rest, the new color does not fill out the whole face as I would like (Possibly because of the subsurf Modifier?). How can I make the whole faces be blue?
You have a bevel between the rim and the interior of your object with one segment. You have given that segment the material of the rim. Because of the rules of Catmull-Clark subdivision, the bevel's face loop will be subdivided, each edge weighted towards pre-existing adjacent edges. Each subdivision receives the material of the original face loop. But the adjacent edges are further away down the length of the interior,so the material is drawn down into it.
If, instead, you have two segments in the bevel, and assign the inner one the internal material, and the outer one the rim material, the subdivisions will stretch the right way.
The top row has a bevel with one segment, the bottom row with two.
Consider creating a second model made with creases.
Consider applying the Subsurf modifier and setting materials for the faces that suit your needs.
In the image above the right faint line is a crease.
The menu location of crease in the image above.
Famous video websites can well demonstrate edge crease use.