If the grouts are different colors?

Do the walls first and let it cure before doing the floor. If your wall grout is cured you will have no trouble wiping off any excess floor grout that might get on the wall.

That's great, but the question is should I gout the floor before I even start to stick tiles to the wall? Also is it necessary to back butter a 13 x 13 tile if it is going on the wall?

Always test a tile before placing another. Put on the thinset.as indicated and then press on a tile. Remove the tile and if the thinset looks good on the back, proceed. No need to backbutter if pattern looks good