A serum-free medium can be more than one grade; for example serum free medium Pro293a is a chemically defined serum-free medium suitable for adherent cells, this serum-free medium is protein-free (no proteins >10kDalton) and is of non animal origin.
If you are currently cultivating your cells with serum and see the benefit of standardizing your cell culture by moving away from serum culturing, you have to follow a procedure called weaning, to adapt your cells to the serum-free environment. The percentage (%) serum added to a cell culture in the range of 5-10% is not conform the in vivo situation, in the human body, the cells are overloaded with food, serum-free is like a healthy diet.
You may have cells, which grow immediately in the new serum-free medium others need the weaning procedure which might takes 4-6 weeks to adapt, depending on cell growth.
One more point of care, when cryopreserving your cells, which are cultured in serum-free medium, avoid using serum, once in serum the weaning procedure is required even when cryopreserving. There is serum-free cryopreservation medium named ProFreeze CD, a chemically defined serum-free freezing medium.