Plywood Sub floor?

1/4 Lauwn plywood and make sure you use level best to fill in any cracks and avoid your new tile's from cracking.

Plywood is not an ideal substrate to lay tile on. You should use a concrete backer board like Dura Rock or Hardiboard over the top of your plywood subfloor.

There's a specific plywood product sold as underlayment for just such an application. It's 1/4" thick and has markings on it to show you where to drive the nails, so you get them spaced properly. Use ring shank or spiral ardox nails. They don't loosen up and pull out.

Cut the pieces carefully, so as not to allow large gaps between sheets.

personally I would refinish the hardwood floor, but thats just me. speak with the people at the lumber yard I would say 3/8 plywood should be ok - NEVER put tiles directly on hardwood flooring it wll shifts with climate changes.

3/4 inch laminit or pressure treated ply wood and make sure you try ot level the floor before laying the new one as best as you can. try to use a laser level or a bubble level and you can also sand the floor to get any rough spots out or high spots this will also help.