I have contour lines, a raster, and a shapefile of the area that needs to be leveled. I have clipped the contour lines and the raster image to the shapefile of the area that needs to be excavated. I was thinking of creating a DEM from the contour lines, but I am not totally sure what to do next.
This is more a civil engineering / grading exercise than a GIS one. I'm not sure what your raster is (imagery? a DEM?) so I will ignore that unless you provide more details.
The first thing you need to do is establish what 'level' is. Do you want it absolutely flat, or with some slope for drainage? Then you need to establish how you want to level the area. Will it be filled in to the highest level, excavated to the lowest level, or balanced in between to minimize the amount of earth that must be removed or brought in from off-site.
Once you have that information determined, you'll need an existing surface and a proposed surface, preferably in DEM format (but you could also use a TIN or Terrain Dataset). GIS can help a bit with evaluating proposed surfaces if trying to balance excavation vs buildup (cut/fill). You can experiment with several surfaces to determine the best balance if you're doing that, or just use the tools to establish how much earth to remove. Depending on if you do this in 3D Analyst or Spatial Analyst, there are various surface and cut/fill tools to use.
We have several questions with more specific information on these tools and the operation your want to do (a cut/fill calculation, even if there is no fill):