Hi Miriam,
Here's how we do it at our cleaning company:
First, you need to assemble the cleaning products you will use. You'll need 2 large buckets 1 for the cleaning solution, and 1 for the rinse water, you'll also need a small portable canister vacuum, or a broom and a dust pan, and then finally get some old towels, rags, or sh-mop to dry the floor.
Use the canister vacuum and vacuum up the dirt and dust. That's right, a vacuum!. Nothing works easier. If you do not have a canister vacuum, use your upright if it has the proper attachments, or resort to the good old broom and dust pan.
Fill the bucket with plain warm water to wash the floor. If the floor is really dirty, add a drop or two of vinegar, or dish soap to 1 gallon of warm water to help cut the grease, or a combination.
Use the mop and scrub the floor, rinsing frequently. Be sure to change both the cleaning and rinse water if it gets too dirty.
After you have cleaned the entire floor, go over the still damp floor with dry towels, rags, or use a sh-mop. You'll be surprised at how much dirt the towels will pick up even after you were certain not a speck of dirt remained. Drying the floor with the towels has the added benefit of removing any water spots that may have left any unsightly spots behind. The floor will shine.
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