Use a high-quality, non-toxic floor cleaner (like Bona) with a soft mop to clean your vinyl floors. You may also use a solution of vinegar and water or any product that is safe for vinyl.
Avoid using any cleaners with wax as they will leave a film on your floors. You also want to avoid using any products with a "shine" agent because that can build up over time, attract dust and dirt, and become very difficult to remove.
Use a vacuum that is suitable for hard surface floors.