Before cleaning the grout, clean the tiles. If you are looking for a fully natural method, you can use vinegar instead of detergent. Try to wipe the areas that do not come off with the hard part of the foam. Mixing the bleach and baking soda is a good idea. If the stains are too resistant, you can leave this mix on the tile for a while. An hour will be enough. After the stain is dissolved, you can rinse the tiles with a clean fabric or cloth.
If you are sure that the tiles are clean, you can start cleaning the grout.