Walking the mop across the floor surface is more effective than
shoving it around.
Sweep the floor before beginning to mop.
If you don't have floor cleaner, dish soap (not dishwashing detergent) works just as well. Use only a small amount diluted in a bucket of warm water.
Treat the dust mop the night before by spraying it down with furniture polish, then sealing it in a plastic bag. The next day the dust mop will be dry and effective.
A dust mop should never leave the floor, though it's tempting to pick it up and shake it. Cover the entire floor with it first, then lift it up and shake it out.
Small commercial mop buckets are available for the home and are far more effective than a regular bucket or the kitchen sink.
The key to thorough floor cleaning is to go over the surface twice. By "walking" the mop across the surface (dragging it rather than shoving it), the floor receives a wetting agent. Flip the mop over and cover the same area to actually clean it.
When the mop has picked up a fair amount of dirt, rinse it out and keep going. Don't attempt to stretch the cleaner--you'll end up just spreading dirt around.