To properly clean a floor wet mopping with a string or microfiber mop will remove the most dirt. It will take extra effort, but having 2 mop buckets and rinsing your mop in clean water before dipping it into the water and cleaning solution will assure you have the cleanest floors.
You will only want to wet mop tile, vinyl, ceramic type floors. it i snot a good idea to wet mop wood and laminate floors because they can buckle. Choose a non-soap based detergent with a medium ph to clean floors. If you want to stick to something natural, vinegar is a good choice. Stay away from mop and go type cleaners that will build up on floors and hold dirt.
Sweep and dust mop/vacuum floors prior to mopping. Be sure to run your broom closely around any trim work and in corners. Dip your mop in the detergent solution and ring very very well. This is very important because a mop that is too wet will cause water to sit on top of the floors and not leave streaks. Make a few passes with the mop, rinse in clean water wring and dip back into detergent solution. Start at one end and work you way to the other apply as much pressure as you can comfortably. For very dirty floors it may take several repetitions to get the floors clean.
Maintaining floors is easier when a coat of wax is applied to vct tile floors. It will also prevent fall risks and improve the life of the floor. Between mopping, maintain by periodically sweeping and dust mopping.