Since I did house cleaning as a business I better know this one. Cleaning marble and ceramic tile floors is a bit tricky in the respect that while detergents & bleach clean well they leave spots and residue when the floor dries, and can damage the grout. So get yourself a good mop and bucket set, one that makes wringing out the mop easy. I always used good cotton mops as some fabric self-wringing mops have plastic or metal parts that can scratch the floors finish. Use very hot water only in cleaning your floors. If you find the mop is to balky and is hard to wring out well, use your scissors and trim to your hearts content.
As for bathtubs, some cleaning products leave bathtubs with a very slippery surface which is dangerous. So to prevent falls try cleaning with softscrub or comet cleaner and a sponge with a scouring pad. This well enable you to scrub away the dirt without damaging the finish of bathtub. Then rinse well with hot water and if you want a shiny look, lightly spray with windex glass cleaner and rub dry with paper towels.