You do not need a check valve as this is a peristaltic pump, it will not siphon back.
As far as tubing, that's a tough question. I personally am just using the "silicone tubing, L/S 13" because it is cheaper. You can get 25 ft for $30 straight from Cole-Parmer. In a perfect world I would probably order the food grade stuff.
Don't run just any tubing in the pump. If you are using this dose chemicals, this is even more important IMO.