How do I remove spots from my new ceramic tiles?
It is likely grout haze.  Wait a week, daily damp mop the grout lines.  Then carefully use a haze remover sparingly (it is an acidic product) The marks will come right up.  Well rinse all areas of any solution, especially any that gets on the grout.  

I made the mistake of not sealing a matte bisque mosaic hex floor tile before grouting early in my tile career.  I then used the haze cleaner too early.  It cleaned the haze alright.  It also removed some color from the grout.  I had to go back and hand stain the grout lines individually.  That's why I'm suggesting a week of curing for your grout.