Fixing these scratches does not ask for much work because only the top protective cover of the hardwood is damaged and the wood underneath is safe. Take a fresh cloth and clean the area free of dust and the grime. You can use a hardwood floor cleaner. After the cleaning, dip a fresh cloth piece in clean water and rinse the area. When the area gets dry, apply a new layer of hardwood protection polish. Do not allow anyone to disturb the spots for a day till the polish gets dry.
For scratches that have passed the protective coat, you will have to do more than polish. Clean the scratch area first with hardwood cleaner and let it dry. Take the steel wool and rub the scratch area in the direction of the grain so that the scratch is removed. Sandpaper can be used for deeper scratches. Rub the wax stick over the scratched area now. The purpose is to blend in the scratch with the surrounding wood. The wax will take ten minutes to set in. After that time, use a clean cloth to buff the surface in back and forth motion.
It is better to call in a professional for deep scratches and grooves. However, if you still want to do it yourself, then get a scouring pad and polyurethane to help with the task. Clean the area first. Moisten the scouring pad with the mineral spirit and press it over the scratches. Rub over the scouring pad and let it dry. Finish up the area with polyurethane.