There's a couple other factors to determine if your joist system is stiff enough for tiles. In addition to 2x8's at 16" o.c. we need to know the species, grade and unsupported span of those 2x8's. You'll probably meet minimums, but lets be safe.
You can keep the old 1x6 subfloor, just repair or replace as necessary. Removing it will create new issues and lots more work cuz you'll have to install blocking under where the new and old meet. You might also compromise the structure since the walls as built on those 1x6's.
After repairing the subfloor, install 1/2" or thicker underlayment grade plywood. Fasten only to the subfloor, not to the joists. NO GLUE.
Install concrete board or matting per instructions, then tiles using the correct thin set mortar. Thin set comes only in powder form in a bag and you'll need it for the tiles and the concrete board too. There's much more of course, but I hope you have a good idea how to proceed.