If you can find one of those scrub brushes with a handle (short, really firm bristles), those are good for scrubbing up stains without having to get down on hands and knees...assuming you have real tiles and not some sort of linoleum tile that would be destroyed by harsh chemicals and scrubbing.
Other than cleaning up stubborn stains, for routine cleaning, a sponge mop works well. Find one that has something on it that looks like a real sponge, not those weird, shiny, non-absorbent spongey plastic looking things that fall apart quickly.
Keep your stringy mop though. They're good for quickly mopping up big spills. Each mop/brush has a good purpose, they just aren't all interchangeable. The stringy ones also work best swished side to side, not pushed back and forth like a scrub brush...I've seen a lot of people have trouble with using mops when they've never used that style before.
If you do need to bleach out a stain that you can't scrub out, I've had good luck with soft scrub. Keep in mind that it's still an abrasive, even if the bottle tries to claim otherwise, so don't use it if an abrasive might damage a surface, but you could use it on areas that are most stained since they're probably already scratched/porous to be holding the stain in the first place (it did a good job on cleaning my grout when I dropped an entire bottle of red wine on my kitchen floor...I can't even blame it on drinking since I hadn't had any yet!)
If you start trying different cleaners, make sure to rinse really well between each one (that stringy mop will be good for rinsing with just clean water).
Another one that works decently for general cleaning, not so much for heavy stains, is either Pinesol or Lysol. Pinesol works a little better, I think, for kitchens because I think it picks up light grease residue better when it settles onto surfaces. A mistake some people make with those types of cleaners with big messes is they think more is better. That'll leave you with a sticky floor if you use too much. It's meant to work best at the recommended dilution on the bottle (though I have used it full strength with a regular kitchen sponge to spot clean a few stubborn areas, and then rinsed a LOT, and it worked well that way too).