I always vacuum first with the roller bar up and just suction.
I dry mop with Libby's 18" long dust mopheads. They're like microfiber fingers and are really good for picking up any dust that the vacuum leaves behind.
For wet mopping, I use Quickie Sponge mops and a bucket of hot water with either dish liquid or mild laundry detergent--depending on what I'm washing. I've never gotten on my hands and knees to wash a floor. Opening and squeezing closed a sponge mop many times seems better for me than wringing out a rag with my hands. It's all about how clean your water is --or isn't.
I do have a Shark steam mop that sprays out a solution and then you steam away.
When dry mopping and steam mopping, I change the pads like a fanatic using several per room. They all can be machine washed and I use Lysol Sanitizing Laundry liquid for them and my bath mats.