Hi! I think the best way to mop the floor is to not use a mop at all. Get some knee pads and wash your floor on your hands and knees.
I have tried so many different mopping systems and none of them work as well as hands & knees cleaning. It doesn't take that much longer and as an added bonus, you can wash your baseboards while your down there. You won't be disappointed!
The easiest way to ensure you're not just moving the dirt around when cleaning floors is to always follow a mopping session with cleaners by a second rinse-mop using clean water only. This will ensure you remove any dirty cleaner residue left on the tile floor and that it is truly clean.