Polishing your fossils is not only a way of presenting a fossil so that it's visually appealing, with all details easily visible. It's also a way to safeguard those same details. By removing surfaces on the fossil face that could easily chip away, you create a smoothed surface that lasts longer and is easier to clean. The polishing isn't difficult, consisting mostly of making successive passes with abrasives to remove scratches and bring a shine to the fossil face. It is time-consuming however, and you'll have to take great care during the polishing process to prevent destroying the very details you're attempting to preserve by grinding them away.
Sand the surface area of the fossil even, removing any depressions or high areas from the surface. Use a sheet of 100-grit wet and dry sandpaper to smooth the surface out, taking care not to remove any of the fossil detail you wish to keep in the process. Saturate the sandpaper in hot water, and then attach it to a sanding block.
Use a figure-8 motion with the fossil across the paper to even the surface out. Rinse the sludge left on the paper regularly and maintain paper wetness until the surface is smooth enough to begin polishing. Rinse off the fossil regularly as well.
Wash the fossil under running water using a soft bristle brush to remove all traces of sanding residue as well as any dirt present on the fossil. Place a bucket under the fossil while washing it to prevent any grit from going down the drains, as this can lead to drain damage.
Remove the scratches from the fossil's surface that were created during the leveling process by sanding the surface with increasingly finer grits of sandpaper. Begin with 200-grit sandpaper, sanding the surface until the roughness created with the 100 grit is smoothed out. Switch to 400, then 800, then 1200-grit sandpapers, using each to pass over the fossil surface removing the scratches left behind by the more coarse paper used before it. The 1200 grit will leave a surface that's smooth and scratch free, ready for the polish.
Create a polish by adding water to 1 tsp. of aluminum oxide polishing compound to create a paste. Spread a small quarter-sized circle of the paste onto the center of a leather pad and then work the paste into the smoothed fossil surface. Vigorously buff the surface of the fossil with the paste until you work up a glossy shine to the surface.
Wash the fossil under running water again after polishing, then dry it off with a lint-free cloth to ready it for display.