Since your data has fewer than 25 values, you should be able to open the Symbology tab for the raster and choose Unique Values. This should allow pretty straight-forward selection of colors, including NOT displaying undesired values just as with vector data. You then have two options to get the symbology to persist.
Option 1: Use a layer file. Right-click the raster in the ToC and choose Save as Layer File. If you add the lyr file to a map instead of the raster, it will symbolize as you want. This method is subject to issues with pathing, but will save your value labels if you set them up to match your cover attribute.
Option 2: Use a color map 'sidecar' file. At the bottom of the Symbology tab click the Colormap drop-down button and choose Export. The colormap (clr) file must have the same name as the raster for it to apply automatically. As long as the clr file is present, when the raster is added it should use those colors. However (at least in my testing) it will not preserve any labels you set up.
You can also manually generate color maps via other means (for instance if your data has more than 25 values), or use the one you have to apply to other rasters using the
Add Colormap GP tool. The help file on Raster Rendering Behavior is where I started my research tree for all of this - it lists default raster rendering behaviors and some options, along with links to other relevant help pages.