Tile floors and grout cleaning
I use a steam mop for tile floors as it's chemical free. I've read not to use vinegar and baking soda on grout as the vinegar is damaging as are waxy cleaners like Murphy's Oil soap or Pine Sol. Rather, use something like Spic and Span or Mr. Clean, said the article I read.
It largely depends on the type of tile you have. Home Depot and Lowes sell various grout and tile cleaning products in the tile department. Most of them are pretty effective but require a lot of scrubbing.
For REALLY tough grime and stains I use muriatic acid. Keep in mind this is a POWERFUL acid, so you need to use a full respirator and thick rubber gloves when working with it. It removes EVERYTHING from grout and tile. You can usually find it in the pool section of most home improvement stores.
If the task is not of heavy dirt I use clorox and baking soda.Sprinkle first baking soda and then pour on the clorox let it sit for 15 minutes. Then scrub and wipe down with a mop and fragrant detergent.
I have used baking soda, hydrogen peroxide and then good old fashioned hands and knees scrubbing. The important part is to give the grout a good seal after it has received all your hard work getting it bright and clean once again. Grout sealer is inexpensive and comes in bottles. I have used old disposable plastic lunch containers, poured the liquid into them and used any paint brush size appropriate for the width of the grout. This usually holds my entry way hall tile for a few years of shoes off policy in place in my home...with effective weekly cleaning in between the deep cleanings.