Remove scuff and shoes marks from the gym floor with a tennis ball attached to a broom or mop handle, or a cleaning agent formulated to remove scuff marks. Ensure it is appropriate for use on gym flooring and is not abrasive.
Sweep the gym floor with a wide broom or a microfiber dust mop.
Mop the gym floor using a cleaning solution of one part white distilled vinegar to two parts warm water--this is more gentle on the finishes of gym floors than cleaners that contain detergents. Microfiber and cotton mops work well for cleaning gym floors.
Clean the floor thoroughly, if it is especially dirty, using an auto-scrubber with a non-abrasive pad. Some gym floors need to be cleaned with an auto-scrubber only a few times per year, while others should be cleaned this way weekly.
Screen, or refinish, the gym floor at least once per year to prevent it from getting slippery. This can be done by a professional cleaning service, or you can do it yourself--it will be a time consuming process, so ensure you have the time and personnel to devote to this task.