You won't believe this but the Mr. Clean Magic Eraser cleans tile floors magically and beautifully! There was an area in front of my door way that was discolored. Since we don't wear shoes in our home, I have a large rubber type mat in front of my door. The tile under the mat was light and the tiles around the mat were much darker. I assumed it was discolored due to the sunlight as I had tried everything to get the tiles clean. Nothing lightened them...that is, until one day I tried the magic eraser on one of the tiles and I couldn't believe it, the tiles got clean. I still can't believe it because my husband went to Home Depot before I tried the magic eraser to buy some strong chemical that is for tile cleaning but the instructions make it clear it's toxic and I have 4 children so I was totally reluctant to want to use that in the home. It took me all day to work on the tiles throughout the entire great room (not just the tiles in front of the doorway) and I used a whole box of erasers but by golly, the tiles came out so clean. As for the grout, not sure what to say about that. I had light grout too but because it's also unsealed, it is dark.
As for a mop, I love the mop I use. It's by Libman and I think it's called the tornado mop. The solution I use varies but I seem drawn to the Mr. Clean varieties. They smell nice and sometimes if I feel like it, I will rinse the floor after I wash it but it also helps if you wash the floor in sections, getting fresh clean water as soon as you see the water turning dark that way you are not washing the floor in dirty water.